corpus.jsonl

{"_id": "0", "title": "The Triangle Inequality", "text": "If $a$ and $b$ are any two real numbers$,$ then \\begin{equation} \\label{eq:1.1.3} |a+b|\\le |a|+|b|. \\end{equation}"}
{"_id": "1", "title": "", "text": "If a nonempty set $S$ of real numbers  is bounded above$,$ then $\\sup S$ is the unique real number $\\beta$ such that \\begin{alist} \\item % (a)  $x\\le\\beta$ for all $x$ in $S;$ \\item % (b)  if $\\epsilon>0$ $($no matter how small$)$$,$ there is an $x_0$ in $S$ such that $x_0> \\beta-\\epsilon.$ \\end{alist}"}
{"_id": "2", "title": "", "text": "If $\\rho$ and $\\epsilon$ are positive$,$ then $n\\epsilon>\\rho$ for some integer $n.$"}
{"_id": "3", "title": "", "text": "The rational numbers are dense in the reals$\\,;$ that is, if $a$ and $b$ are real numbers with $a<b,$ there is a rational number $p/q$ such that $a<p/q<b$."}
{"_id": "4", "title": "", "text": "The set of irrational numbers is dense in the reals$\\,;$ that is, if $a$ and $b$ are real numbers with $a<b,$  there is an irrational number $t$ such that $a<t<b.$"}
{"_id": "5", "title": "", "text": "If a nonempty set $S$ of real numbers  is bounded below$,$ then $\\inf S$ is the unique real number $\\alpha$ such that \\begin{alist} \\item % \\part{a}  $x\\ge\\alpha$ for all $x$ in $S;$ \\item % (b)  if $\\epsilon>0$ $($no matter how small$\\,)$, there is an $x_0$ in $S$ such that $x_0< \\alpha+\\epsilon.$ \\end{alist}"}
{"_id": "6", "title": "Principle of Mathematical Induction", "text": "Let $P_1,$ $P_2, $\\dots$,$ $P_n,$ \\dots\\ be propositions$,$ one for each positive integer$,$ such that \\begin{alist} \\item % (a)  $P_1$ is true$;$ \\item % (b)  for each positive integer $n,$  $P_n$  implies $P_{n+1}.$ \\end{alist} Then $P_n$ is true for each positive integer $n.$"}
{"_id": "7", "title": "", "text": "Let $n_0$ be any integer $($positive$,$  negative$,$ or zero$)$$.$ Let $P_{n_0},$ $P_{n_0+1},$ \\dots$,$ $P_n,$ \\dots\\ be propositions$,$  one for each integer $n\\ge n_0,$ such that \\begin{alist} \\item % (a)  $P_{n_0}$ is true$\\,;$ \\item % (b)  for each integer $n\\ge n_0,$ $P_n$  implies $P_{n+1}.$ \\end{alist} Then $P_n$ is true for every integer  $n\\ge n_0.$"}
{"_id": "8", "title": "", "text": "Let $n_0$ be any integer $($positive$,$  negative$,$ or zero$)$$.$ Let $P_{n_0},$ $P_{n_0+1}, $\\dots$,$ $P_n,$ \\dots\\ be propositions$,$  one for each integer $n\\ge n_0,$ such that \\begin{alist} \\item % (a)  $P_{n_0}$ is true$\\,;$ \\item % (b) for $n\\ge n_0,$ $P_{n+1}$ is true if $P_{n_0},$ $P_{n_0+1}, $\\dots$,$ $P_n$ are all true. \\end{alist} Then $P_n$ is true for $n\\ge n_0.$"}
{"_id": "9", "title": "", "text": "\\begin{alist} \\item % (a)   The union of open sets is open$.$ \\item % (b)  The intersection of closed sets is closed$.$ \\end{alist} These statements apply to arbitrary collections, finite or infinite, of open and closed sets$.$"}
{"_id": "10", "title": "", "text": "no point of $S^c$ is a limit point of~$S.$"}
{"_id": "11", "title": "", "text": "If ${\\mathcal H}$ is an open covering of a closed and bounded subset $S$ of the real line$,$ then $S$ has an open covering $\\widetilde{\\mathcal H}$ consisting of finitely many open sets belonging to ${\\mathcal H}.$"}
{"_id": "12", "title": "", "text": "Every bounded infinite set of real numbers has at least one limit point$.$"}
{"_id": "13", "title": "", "text": "then it is unique$\\,;$ that is$,$ if \\begin{equation} \\label{eq:2.1.7} \\lim_{x\\to x_0} f(x)=L_1\\mbox{\\quad and \\quad}\\lim_{x\\to x_0} f(x)= L_2, \\end{equation} then $L_1=L_2.$"}
{"_id": "14", "title": "", "text": "\\begin{equation}\\label{eq:2.1.9} \\lim_{x\\to x_0} f(x)=L_1\\mbox{\\quad and \\quad}\\lim_{x\\to x_0} g(x)= L_2, \\end{equation} then \\begin{eqnarray} \\lim_{x\\to x_0} (f+g)(x)\\ar= L_1+L_2,\\label{eq:2.1.10}\\\\ \\lim_{x\\to x_0} (f-g)(x)\\ar= L_1-L_2,\\label{eq:2.1.11}\\\\ \\lim_{x\\to x_0} (fg)(x)\\ar= L_1L_2,\\label{eq:2.1.12}\\\\ \\arraytext{and, if $L_2\\ne0$,}\\\\ \\lim_{x\\to x_0}\\left(\\frac{f}{g}\\right)(x)\\ar= \\frac{L_1}{ L_2}.\\label{eq:2.1.13} \\end{eqnarray}"}
{"_id": "15", "title": "", "text": "A function $f$ has a limit at $x_0$ if and only if it has left- and right-hand limits at $x_0,$ and they are equal.  More specifically$,$ $$ \\lim_{x\\to x_0} f(x)=L $$ if and only if $$ f(x_0+)=f(x_0-)=L. $$"}
{"_id": "16", "title": "", "text": "Suppose that $f$ is monotonic on $(a,b)$ and define $$ \\alpha=\\inf_{a<x<b}f(x) \\mbox{\\quad and \\quad} \\beta=\\sup_{a<x<b}f(x). $$ \\begin{alist} \\item % (a)  If $f$ is nondecreasing$,$ then $f(a+)=\\alpha$ and  $f(b-)=\\beta.$ \\item % (b) If $f$ is nonincreasing$,$ then $f(a+)=\\beta$ and $f(b-)=\\alpha.$ \\\\ $($Here $a+=-\\infty$ if $a=-\\infty$ and $b-=\\infty$ if $b=\\infty.)$ \\item %  (c)  If $a<x_0<b$, then $f(x_0+)$ and $f(x_0-)$ exist and are finite$\\,;$ moreover$,$ $$ f(x_0-)\\le f(x_0)\\le f(x_0+) $$  if $f$ is nondecreasing$,$ and $$ f(x_0-)\\ge f(x_0)\\ge f(x_0+) $$  if $f$ is nonincreasing$.$ \\end{alist}"}
{"_id": "17", "title": "", "text": "If $f$ is bounded on $[a,x_0),$ then $\\beta=\\limsup_{x\\to x_0-}f(x)$ exists and is the unique real number with the following properties$\\,:$ \\begin{alist} \\item % (a) If $\\epsilon>0$, there is an $a_1$ in $[a,x_0)$ such that \\begin{equation} \\label{eq:2.1.22} f(x)<\\beta+\\epsilon\\mbox{\\quad if \\quad} a_1\\le x<x_0. \\end{equation} \\item % (b) If $\\epsilon>0$ and $a_1$ is in $[a,x_0),$ then $$ f(\\overline x)>\\beta-\\epsilon\\mbox{\\quad for some }\\overline x\\in[a_1,x_0). $$ \\end{alist}"}
{"_id": "18", "title": "", "text": "If $f$ is bounded on $[a,x_0),$ then $\\alpha=\\liminf_{x\\to x_0-}f(x)$ exists and is the unique real number with the following properties: \\begin{alist} \\item % (a) If $\\epsilon>0,$ there is an $a_1$ in $[a,x_0)$ such that $$ f(x)>\\alpha-\\epsilon\\mbox{\\quad if \\quad} a_1\\le x<x_0. $$ \\item % (b) If $\\epsilon>0$ and $a_1$ is in $[a,x_0),$ then $$ f(\\overline x)<\\alpha+\\epsilon\\mbox{\\quad for some }\\overline x\\in[a_1,x_0). $$ \\end{alist}"}
{"_id": "19", "title": "", "text": "\\vspace*{6pt} \\begin{alist} \\item % (a) A function $f$ is continuous at $x_0$ if and only if $f$ is defined on an open interval $(a,b)$ containing $x_0$ and for each $\\epsilon>0$ there is a $\\delta >0$ such that \\begin{equation}\\label{eq:2.2.1} |f(x)-f(x_0)|<\\epsilon \\end{equation} whenever $|x-x_0|<\\delta.$ \\item % (b) A function $f$ is continuous from the right at $x_0$ if and only if $f$ is defined on an interval $[x_0,b)$ and for each $\\epsilon> 0$ there is a $\\delta>0$ such that $\\eqref{eq:2.2.1}$ holds whenever $x_0\\le x<x_0+ \\delta.$ \\item % (c) A function $f$ is continuous from the left at $x_0$ if and only if $f$ is defined on an interval $(a,x_0]$ and for each $\\epsilon >0$ there is a $\\delta>0$ such that $\\eqref{eq:2.2.1}$ holds whenever $x_0-\\delta<x\\le x_0.$ \\end{alist}"}
{"_id": "20", "title": "", "text": "If $f$ and $g$ are continuous on a set $S,$ then so are $f+g,$ $f-g,$ and $fg.$ In addition$,$ $f/g$ is continuous at each $x_0$ in $S$ such that $g(x_0)\\ne0.$"}
{"_id": "21", "title": "", "text": "Suppose that $g$ is continuous at $x_0,$ $g(x_0)$ is an interior point of $D_f,$ and $f$ is continuous at $g(x_0).$ Then  $f\\circ g$ is continuous at $x_0.$"}
{"_id": "22", "title": "", "text": "If $f$ is continuous on a finite closed interval $[a,b],$ then $f$ is bounded on $[a,b].$"}
{"_id": "23", "title": "", "text": "Suppose that $f$ is continuous on a finite closed interval $[a,b].$ Let $$ \\alpha=\\inf_{a\\le x\\le b}f(x)\\mbox{\\quad and \\quad}\\beta=\\sup_{a\\le x\\le b}f(x). $$ Then $\\alpha$ and $\\beta$ are respectively the minimum and maximum of $f$ on $[a,b];$ that is$,$  there are points $x_1$ and $x_2$ in $[a,b]$ such that $$ f(x_1)=\\alpha\\mbox{\\quad and \\quad} f(x_2)=\\beta. $$"}
{"_id": "24", "title": "Intermediate Value Theorem", "text": "Suppose that $f$ is continuous on $[a,b],$ $f(a)\\ne f(b),$ and $\\mu$ is between $f(a)$ and $f(b).$ Then $f(c)=\\mu$ for some $c$ in $(a,b).$"}
{"_id": "25", "title": "", "text": "If $f$ is continuous on a closed and bounded interval $[a,b],$ then $f$ is uniformly continuous on $[a,b].$"}
{"_id": "26", "title": "", "text": "If $f$ is monotonic and nonconstant on $[a,b],$ then $f$ is continuous on $[a,b]$ if and only if its range $R_f=\\set{f(x)}{x\\in[a,b]}$ is the closed interval with endpoints $f(a)$ and $f(b).$"}
{"_id": "27", "title": "", "text": "Suppose that $f$ is increasing and continuous on $[a,b],$ and let $f(a)=c$ and $f(b)=d.$ Then there is a unique function $g$ defined on $[c,d]$ such that \\begin{equation}\\label{eq:2.2.17} g(f(x))=x,\\quad a\\le x\\le b, \\end{equation} and \\begin{equation}\\label{eq:2.2.18} f(g(y))=y,\\quad c\\le y\\le d. \\end{equation} Moreover$,$ $g$ is continuous and increasing on $[c,d].$"}
{"_id": "28", "title": "", "text": "If $f$ is differentiable at $x_0,$ then $f$ is continuous at $x_0.$"}
{"_id": "29", "title": "", "text": "If $f$ and $g$ are differentiable at $x_0,$ then so are $f+g,$ $f-g,$ and $fg,$ with \\begin{alist} \\item % (a)  $(f+g)'(x_0)=f'(x_0)+g'(x_0);$ \\item % (b)  $(f-g)'(x_0)=f'(x_0)-g(x_0);$ \\item % (c)  $(fg)'(x_0)=f'(x_0)g(x_0)+f(x_0)g'(x_0).$ \\end{alist}  The quotient $f/g$ is differentiable at $x_0$ if $g(x_0)\\ne0,$ with \\begin{alist} \\setcounter{lcal}{3} \\item % (d) $\\dst{\\left(\\frac{f}{g}\\right)' (x_0)= \\frac{f'(x_0)g(x_0)-f(x_0)g'(x_0)}{\\left[g(x_0)\\right]^2}}.$ \\end{alist}"}
{"_id": "30", "title": "The Chain Rule", "text": "Suppose that  $g$ is differentiable at $x_0$ and $f$ is differentiable at $g(x_0).$ Then the composite function $h=f\\circ g,$ defined by $$ h(x)=f(g(x)), $$ is differentiable at $x_0,$ with $$ h'(x_0)=f'(g(x_0))g'(x_0). $$"}
{"_id": "31", "title": "", "text": "If $f$ is differentiable at a local extreme point $x_0\\in D_{f}^{0},$ then $f'(x_0)=~0.$"}
{"_id": "32", "title": "", "text": "Suppose that  $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b),$ and $f(a)=f(b).$ Then $f'(c)=0$ for some $c$ in the open interval $(a,b).$"}
{"_id": "33", "title": "Intermediate Value Theorem for Derivatives", "text": "Suppose that  $f$ is differentiable on $[a,b],$  $f'(a)\\ne f'(b),$ and $\\mu$ is between $f'(a)$ and $f'(b).$ Then $f'(c)=\\mu$ for some $c$ in $(a,b).$"}
{"_id": "34", "title": "Generalized Mean Value Theorem", "text": "If $f$ and $g$ are continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b),$ then \\begin{equation} \\label{eq:2.3.20} [g(b)-g(a)]f'(c)=[f(b)-f(a)]g'(c) \\end{equation} for some $c$ in $(a,b).$"}
{"_id": "35", "title": "Mean Value Theorem", "text": "If $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b),$ then $$ f'(c)=\\frac{f(b)-f(a)}{ b-a} $$ for some $c$ in $(a,b).$"}
{"_id": "36", "title": "", "text": "If $f'(x)=0$ for all $x$ in $(a,b),$ then $f$ is constant on $(a,b).$"}
{"_id": "37", "title": "", "text": "If $f'$ exists and does not change sign on $(a,b),$ then $f$ is monotonic on $(a,b):$ increasing$,$ nondecreasing$,$ decreasing$,$ or nonincreasing as $$ f'(x)>0,\\quad f'(x)\\ge0,\\quad f'(x)<0,\\mbox{\\quad or\\quad} f'(x) \\le0, $$ respectively$,$ for all $x$ in $(a,b).$"}
{"_id": "38", "title": "", "text": "If $$ |f'(x)|\\le M,\\quad a<x<b, $$ then \\begin{equation} \\label{eq:2.3.21} |f(x)-f(x')|\\le M |x-x'|,\\quad x, x'\\in (a,b). \\end{equation}"}
{"_id": "39", "title": "", "text": "Suppose that $f$ and $g$ are differentiable and $g'$ has no zeros on $(a,b).$ Let \\begin{equation}\\label{eq:2.4.1} \\lim_{x\\to b-}f(x)=\\lim_{x\\to b-}g(x)=0 \\end{equation} \\newpage \\noindent or \\begin{equation}\\label{eq:2.4.2} \\lim_{x\\to b-}f(x)=\\pm\\infty\\mbox{\\quad and \\quad} \\lim_{x\\to b-}g(x)=\\pm\\infty, \\end{equation} and suppose that \\begin{equation}\\label{eq:2.4.3} \\lim_{x\\to b-}\\frac{f'(x)}{ g'(x)}=L\\quad\\mbox{$($finite or $\\pm \\infty)$}. \\end{equation} Then \\begin{equation}\\label{eq:2.4.4} \\lim_{x\\to b-}\\frac{f(x)}{ g(x)}=L. \\end{equation}"}
{"_id": "40", "title": "", "text": "If $f^{(n)}(x_0)$ exists for some integer $n\\ge1$ and $T_n$ is the $n$th Taylor polynomial of $f$ about $x_0,$ then \\begin{equation}\\label{eq:2.5.5} \\lim_{x\\to x_0}\\frac{f(x)-T_n(x)}{(x-x_0)^n}=0. \\end{equation}"}
{"_id": "41", "title": "", "text": "Suppose that $f$ has $n$ derivatives at $x_0$ and $n$ is the smallest positive integer such that $f^{(n)}(x_0)\\ne 0.$ \\begin{alist} \\item % (a) If $n$ is odd$,$ $x_0$ is not a local extreme point of $f.$ \\item % (b)  If $n$ is even$,$ $x_0$ is a local maximum of $f$ if $f^{(n)}(x_0)<0,$ or a local mininum of $f$ if $f^{(n)}(x_0)>0.$ \\end{alist}"}
{"_id": "42", "title": "Taylor's Theorem", "text": "Suppose that $f^{(n+1)}$ exists on an open interval $I$ about $x_0,$ and let $x$ be in $I.$ Then the remainder $$ R_n(x)=f(x)-T_n(x) $$ can be written as $$ R_n(x)=\\frac{f^{(n+1)}(c)}{(n+1)!}(x-x_0)^{n+1}, $$ where $c$ depends upon $x$ and is between $x$ and $x_0.$"}
{"_id": "43", "title": "Extended Mean Value Theorem", "text": "Suppose that $f$ is continuous on a finite closed interval $I$ with endpoints $a$ and $b$ $($that is, either $I=(a,b)$ or $I=(b,a)),$ $f^{(n+1)}$ exists on the open interval $I^0,$ and$,$ if $n>0,$ that $f'$, \\dots, $f^{(n)}$ exist and are continuous at $a.$ Then \\begin{equation}\\label{eq:2.5.17} f(b)-\\sum_{r=0}^n\\frac{f^{(r)}(a)}{ r!}(b-a)^r=\\frac{f^{(n+1)}(c)}{(n+1)!} (b-a)^{n+1} \\end{equation} for some $c$ in $I^0.$"}
{"_id": "44", "title": "", "text": "If $f$ is unbounded on $[a,b],$ then $f$ is not integrable on $[a,b].$"}
{"_id": "45", "title": "", "text": "Let $f$ be bounded on $[a,b]$,  and let $P$ be a partition of $[a,b].$  Then \\begin{alist} \\item % (a)  The upper sum $S(P)$ of $f$ over $P$ is the supremum  of the set of all Riemann sums of $f$ over $P.$ \\item % (b)  The lower sum $s(P)$ of $f$ over $P$ is the infimum  of the set of all Riemann sums of $f$ over $P.$ \\end{alist}"}
{"_id": "46", "title": "", "text": "If $f$ is bounded on $[a,b],$ then \\begin{equation} \\label{eq:3.2.6} \\underline{\\int_a^b}f(x)\\,dx\\le\\overline{\\int_a^b}f(x)\\,dx. \\end{equation}"}
{"_id": "47", "title": "", "text": "If $f$ is integrable on $[a,b],$ then $$ \\underline{\\int_a^b}f(x)\\,dx=\\overline{\\int_a^b}f(x)\\,dx=\\int_a^b f(x)\\,dx. $$"}
{"_id": "48", "title": "", "text": "If $f$ is bounded on $[a,b]$ and \\begin{equation} \\label{eq:3.2.16} \\underline{\\int_a^b} f(x)\\,dx=\\overline{\\int_a^b}f(x)\\,dx=L, \\end{equation} then $f$ is integrable on $[a,b]$ and \\begin{equation} \\label{eq:3.2.17} \\int_a^b f(x)\\,dx=L. \\end{equation}"}
{"_id": "49", "title": "", "text": "A bounded function $f$ is integrable on $[a,b]$ if and only if $$ \\underline{\\int_a^b}f(x)\\,dx=\\overline{\\int_a^b}f(x)\\,dx. $$"}
{"_id": "50", "title": "", "text": "If $f$ is bounded on $[a,b],$ then $f$ is integrable on $[a,b]$  if and only if for each $\\epsilon>0$ there is a partition $P$ of $[a,b]$ for which \\begin{equation} \\label{eq:3.2.19} S(P)-s(P)<\\epsilon. \\end{equation}"}
{"_id": "51", "title": "", "text": "If $f$ is continuous on $[a,b],$ then $f$ is integrable on $[a,b]$."}
{"_id": "52", "title": "", "text": "If $f$ is monotonic on $[a,b],$ then $f$ is integrable on $[a,b]$."}
{"_id": "53", "title": "", "text": "If $f$ and $g$ are integrable on $[a,b],$ then so is $f+g,$ and \\vskip4pt $$ \\int_a^b (f+g)(x)\\,dx=\\int_a^b f(x)\\,dx+\\int_a^b g(x)\\,dx. $$"}
{"_id": "54", "title": "", "text": "If $f$ is integrable on $[a,b]$ and $c$ is a constant$,$ then $cf$ is integrable on $[a,b]$ and $$ \\int_a^b cf(x)\\,dx=c\\int_a^b f(x)\\,dx. $$"}
{"_id": "55", "title": "", "text": "If $f_1,$ $f_2,$ \\dots$,$ $f_n$ are integrable on $[a,b]$ and $c_1,$ $c_2,$ \\dots$,$ $c_n$ are constants$,$ then $c_1f_1+c_2f_2+\\cdots+ c_nf_n$ is integrable on $[a,b]$ and \\begin{eqnarray*} \\int_a^b (c_1f_1+c_2f_2+\\cdots+c_nf_n)(x)\\,dx\\ar=c_1\\int_a^b f_1(x)\\,dx +c_2\\int_a^b f_2(x)\\,dx\\\\ \\ar{}+\\cdots+c_n\\int_a^b f_n(x)\\,dx. \\end{eqnarray*}"}
{"_id": "56", "title": "", "text": "If $f$ and $g$ are integrable on $[a,b]$ and $f(x)\\le g(x)$ for $a\\le x\\le b,$ then \\begin{equation}\\label{eq:3.3.1} \\int_a^b f(x)\\,dx\\le\\int_a^b g(x)\\,dx. \\end{equation}"}
{"_id": "57", "title": "", "text": "If $f$ is integrable on $[a,b],$ then so is $|f|$, and \\begin{equation} \\label{eq:3.3.3} \\left|\\int_a^b f(x)\\,dx\\right|\\le\\int_a^b |f(x)|\\,dx. \\end{equation}"}
{"_id": "58", "title": "", "text": "If $f$ and $g$ are integrable on $[a,b],$ then so is the product $fg.$"}
{"_id": "59", "title": "First Mean Value Theorem for Integrals", "text": "Suppose that $u$ is continuous and $v$ is integrable and nonnegative on $[a,b].$ Then \\begin{equation} \\label{eq:3.3.8} \\int_a^b u(x)v(x)\\,dx=u(c)\\int_a^b v(x)\\,dx \\end{equation} for some $c$ in $[a,b]$."}
{"_id": "60", "title": "", "text": "If $f$ is integrable on $[a,b]$ and $a\\le a_1<b_1\\le b,$ then $f$ is integrable on $[a_1,b_1].$"}
{"_id": "61", "title": "", "text": "If $f$ is integrable on $[a,b]$ and $[b,c],$ then $f$ is integrable on $[a,c],$ and \\begin{equation} \\label{eq:3.3.12} \\int_a^cf(x)\\,dx=\\int_a^bf(x)\\,dx+\\int_b^cf(x)\\,dx. \\end{equation}"}
{"_id": "62", "title": "", "text": "If $f$ is integrable on $[a,b]$ and $a\\le c\\le b,$ then the function $F$ defined by $$  F(x)=\\int_c^x f(t)\\,dt $$  satisfies a Lipschitz condition on $[a,b],$ and is therefore continuous on $[a,b].$"}
{"_id": "63", "title": "", "text": "If $f$ is integrable on $[a,b]$ and $a\\le c\\le b,$ then $F(x)=\\int_c^x f(t)\\,dt$ is differentiable at any point $x_0$ in $(a,b)$ where $f$ is continuous$,$ with $F'(x_0)=f(x_0).$ If $f$ is continuous from the right at $a,$ then $F_+'(a)=f(a)$. If $f$ is continuous from the left at $b,$ then $F_-'(b)=f(b).$"}
{"_id": "64", "title": "", "text": "Suppose that $F$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b),$ and $f$ is integrable on $[a,b].$ Suppose  also that $$ F'(x)=f(x),\\quad a<x<b. $$ Then \\begin{equation}\\label{eq:3.3.14} \\int_a^b f(x)\\,dx=F(b)-F(a). \\end{equation}"}
{"_id": "65", "title": "Fundamental Theorem of  Calculus", "text": "If $f$ is continuous on $[a,b],$ then $f$ has an antiderivative on $[a,b].$ Moreover$,$ if $F$ is any antiderivative of $f$ on $[a,b],$ then $$ \\int_a^b f(x)\\,dx=F(b)-F(a). $$"}
{"_id": "66", "title": "Integration by Parts", "text": "If $u'$ and $v'$ are integrable on $[a,b],$ then \\begin{equation}\\label{eq:3.3.16} \\int_a^b u(x)v'(x)\\,dx=u(x)v(x)\\bigg|^b_a-\\int_a^b v(x)u'(x)\\,dx. \\end{equation}"}
{"_id": "67", "title": "Second Mean Value Theorem for Integrals", "text": "Suppose that $f'$ is nonnegative and integrable and $g$ is continuous on $[a,b].$  Then \\begin{equation}\\label{eq:3.3.17} \\int_a^b f(x)g(x)\\,dx=f(a)\\int_a^c g(x)\\,dx+f(b)\\int_c^b g(x)\\,dx \\end{equation} for some $c$ in $[a,b].$"}
{"_id": "68", "title": "", "text": "Suppose that the transformation $x=\\phi(t)$ maps the interval $c\\le t\\le d$ into the interval $a\\le x\\le b,$ with $\\phi(c)=\\alpha$ and $\\phi(d)=\\beta,$ and let $f$ be continuous on $[a,b].$ Let $\\phi'$ be integrable on $[c,d].$ Then \\begin{equation}\\label{eq:3.3.21} \\int_\\alpha^\\beta f(x)\\,dx=\\int_c^d f(\\phi(t))\\phi'(t)\\,dt. \\end{equation}"}
{"_id": "69", "title": "", "text": "Suppose that $\\phi'$ is integrable and $\\phi$ is monotonic on $[c,d],$ and the transformation $x=\\phi(t)$ maps $[c,d]$ onto $[a,b].$ Let $f$ be bounded on $[a,b].$ Then $$ g(t)=f(\\phi(t))\\phi'(t) $$ is integrable on $[c,d]$ if and only if $f$ is integrable over $[a,b],$ and in this case $$ \\int_a^b f(x)\\,dx=\\int_c^d f(\\phi(t))|\\phi'(t)|\\,dt. $$"}
{"_id": "70", "title": "", "text": "Suppose that   $f_1,$ $f_2,$ \\dots$,$ $f_n$  are locally integrable on $[a,b)$ and that $\\int_a^bf_1(x)\\,dx,$ $\\int_a^bf_2(x)\\,dx,$ \\dots$,$ $\\int_a^bf_n(x)\\,dx$ converge$.$ Let $c_1,$ $c_2,$ \\dots$,$ $c_n$ be constants$.$ Then  $\\int_a^b(c_1f+c_2f_1+\\cdots+c_nf_n)(x)\\,dx$ converges and \\begin{eqnarray*} \\int_a^b (c_1f_1+c_2f_2+\\cdots+c_nf_n)(x)\\,dx\\ar=c_1\\int_a^b f_1(x)\\,dx +c_2\\int_a^b f_2(x)\\,dx\\\\ \\ar{}+\\cdots+c_n\\int_a^b f_n(x)\\,dx. \\end{eqnarray*}"}
{"_id": "71", "title": "", "text": "If $f$ is nonnegative and locally integrable on $[a,b),$ then $\\int_a^b f(x)\\,dx$ converges if the function $$ F(x)=\\int_a^x f(t)\\,dt $$ is bounded on $[a,b)$, and $\\int_a^b f(x)\\,dx=\\infty$ if it is not. These are the only possibilities, and $$ \\int_a^b f(t)\\,dt=\\sup_{a\\le x<b}F(x) $$ in either case$.$"}
{"_id": "72", "title": "Comparison Test", "text": "If $f$ and $g$ are locally integrable on $[a,b)$ and \\begin{equation}\\label{eq:3.4.2} 0\\le f(x)\\le g(x),\\quad a\\le x<b, \\end{equation} then \\vskip3pt \\noindent \\part{a}\\phantom{xxxxxxxxxxxxxxxxxxxx} $\\dst\\int_a^b f(x)\\,dx<\\infty \\mbox{\\quad if\\quad}\\dst\\int_a^b g(x)\\,dx<\\infty$ \\\\ \\vskip3pt \\noindent and\\\\ \\vskip3pt \\noindent \\part{b}\\phantom{xxxxxxxxxxxxxxxxxxxx} $\\dst\\int_a^b g(x)\\,dx= \\infty\\mbox{\\quad if\\quad}\\dst\\int_a^b f(x)\\,dx=\\infty$."}
{"_id": "73", "title": "", "text": "Suppose that  $f$ and $g$ are locally integrable on $[a,b),$ $g(x)>0$ and $f(x)\\ge0$ on some subinterval $[a_1,b)$ of $[a,b),$ and \\begin{equation}\\label{eq:3.4.3} \\lim_{x\\to b-}\\frac{f(x)}{ g(x)}=M. \\end{equation} \\begin{alist} \\item % (a) If $0<M<\\infty,$ then $\\int_a^b f(x)\\,dx$ and $\\int_a^b g(x)\\,dx$ converge or diverge together. \\item % (b)  If $M=\\infty$ and $\\int_a^b g(x)\\,dx=\\infty,$ then $\\int_a^b f(x)\\,dx=\\infty$. \\item % (c)  If $M=0$ and $\\int_a^b g(x)\\,dx<\\infty,$ then $\\int_a^b f(x)\\,dx<\\infty$. \\end{alist}"}
{"_id": "74", "title": "", "text": "If $f$ is locally integrable on $[a,b)$ and $\\int_a^b |f(x)|\\,dx<\\infty,$ then $\\int_a^b f(x)\\,dx$ converges$;$ that is$,$ an absolutely convergent integral is convergent$.$"}
{"_id": "75", "title": "", "text": "Suppose that  $f$ is continuous  and its antiderivative $F(x)=\\int_a^x f(t)\\,dt$ is bounded on $[a,b).$ Let $g'$ be absolutely integrable on $[a,b),$ and   suppose that \\begin{equation}\\label{eq:3.4.9} \\lim_{x\\to b-} g(x)=0. \\end{equation} Then $\\int_a^b f(x)g(x)\\,dx$ converges$.$"}
{"_id": "76", "title": "", "text": "Suppose that  $u$ is continuous on $[a,b)$ and $\\int_a^bu(x)\\,dx$   diverges$.$ Let $v$ be positive and differentiable on $[a,b),$ and  suppose that  $\\lim_{x\\to b-}v(x)=\\infty$ and $v'/v^2$ is absolutely integrable on $[a,b).$ Then $\\int_a^b u(x)v(x)\\,dx$  diverges$.$"}
{"_id": "77", "title": "", "text": "Suppose that  $g$ is monotonic on $[a,b)$ and $\\int_a^b g(x)\\,dx=\\infty.$ Let $f$ be locally integrable on $[a,b)$ and $$ \\int_{x_j}^{x_{j+1}} |f(x)|\\,dx\\ge\\rho,\\quad j\\ge0, $$ for some positive $\\rho,$ where $\\{x_j\\}$ is an increasing infinite sequence of points in $[a,b)$ such that $\\lim_{j\\to\\infty}x_j=b$ and $x_{j+1}-x_j\\le M,$ $j\\ge0,$ for some $M.$ Then $$ \\int_a^b |f(x)g(x)|\\,dx=\\infty. $$"}
{"_id": "78", "title": "", "text": "Suppose that  $\\phi$ is monotonic and $\\phi'$ is locally integrable on either of the half-open intervals $I=[c,d)$ or $(c,d],$ and let $x=\\phi(t)$ map $I$ onto either of the half-open intervals $J=[a,b)$ or $J=(a,b].$ Let $f$ be locally integrable on $J.$ Then the improper integrals $$ \\int_a^b f(x)\\,dx\\mbox{\\quad and\\quad}\\int_c^d f\\left(\\phi(t)\\right)|\\phi'(t)|\\,dt $$ \\newpage \\noindent diverge or converge together$,$ in the latter case to the same value. The same conclusion holds if $\\phi$ and $\\phi'$ have the stated properties only on the open interval $(a,b),$ the transformation $x=\\phi(t)$ maps $(c,d)$ onto $(a,b),$ and $f$ is locally integrable on $(a,b).$"}
{"_id": "79", "title": "", "text": "Let $f$ be defined on $[a,b].$ Then $f$ is continuous at $x_0$ in $[a,b]$ if and only if $w_f(x_0)=0.$ $($Continuity at $a$ or $b$ means continuity from the right or left, respectively.$)$"}
{"_id": "80", "title": "", "text": "A bounded function $f$ is integrable on a finite interval $[a,b]$ if and only if the set $S$ of discontinuities of $f$ in $[a,b]$ is of Lebesgue measure zero$.$"}
{"_id": "81", "title": "", "text": "The limit of a convergent sequence is unique$.$"}
{"_id": "82", "title": "", "text": "A convergent sequence is bounded$.$"}
{"_id": "83", "title": "", "text": "\\begin{alist} \\item % (a)  If $\\{s_n\\}$ is nondecreasing$,$ then $\\lim_{n\\to\\infty}s_n=\\sup\\{s_n\\}.$ \\item % (b If $\\{s_n\\}$ is nonincreasing$,$ then $\\lim_{n\\to\\infty}s_n= \\inf\\{s_n\\}.$ \\end{alist}"}
{"_id": "84", "title": "", "text": "Let $\\lim_{x\\to\\infty} f(x)=L,$ where $L$ is  in the extended reals$,$ and suppose that $s_n=f(n)$ for large $n.$  Then $$ \\lim_{n\\to\\infty}s_n=L. $$"}
{"_id": "85", "title": "", "text": "Let \\begin{equation}\\label{eq:4.1.4} \\lim_{n\\to\\infty} s_n=s\\mbox{\\quad and\\quad}\\lim_{n\\to\\infty} t_n=t, \\end{equation} where $s$ and $t$ are finite$.$  Then \\begin{equation}\\label{eq:4.1.5} \\lim_{n\\to\\infty} (cs_n)=cs \\end{equation} if $c$ is a constant$;$ \\begin{eqnarray} \\lim_{n\\to\\infty}(s_n+t_n)\\ar=s+t,\\label{eq:4.1.6}\\\\ \\lim_{n\\to\\infty}(s_n-t_n)\\ar=s-t, \\label{eq:4.1.7}\\\\ \\lim_{n\\to\\infty}(s_nt_n)\\ar=st,\\label{eq:4.1.8}\\\\ \\arraytext{and}\\nonumber\\\\ \\lim_{n\\to\\infty}\\frac{s_n}{ t_n}\\ar=\\frac{s}{ t}\\label{eq:4.1.9} \\end{eqnarray} if $t_n$ is nonzero for all $n$ and $t\\ne0$."}
{"_id": "86", "title": "", "text": "\\begin{alist} \\item % (a) If $\\{s_n\\}$ is bounded above and does not diverge to $-\\infty,$ then there is a unique real number $\\overline{s}$ such that$,$ if $\\epsilon>0,$ \\begin{equation}\\label{eq:4.1.16} s_n<\\overline{s}+\\epsilon\\mbox{\\quad for large $n$} \\end{equation} and \\begin{equation}\\label{eq:4.1.17} s_n>\\overline{s}-\\epsilon\\mbox{\\quad for infinitely many  $n$}. \\end{equation} \\item % (b) If $\\{s_n\\}$ is bounded below and does not diverge to $\\infty,$ then there is a unique real number $\\underline{s}$ such that$,$ if $\\epsilon >0,$ \\begin{equation}\\label{eq:4.1.18} s_n>\\underline{s}-\\epsilon\\mbox{\\quad for large $n$} \\end{equation} and \\begin{equation}\\label{eq:4.1.19} s_n<\\underline{s}+\\epsilon\\mbox{\\quad for infinitely many $n$}. \\end{equation} \\end{alist}"}
{"_id": "87", "title": "", "text": "Every sequence $\\{s_n\\}$ of real numbers has a unique limit superior$,$ $\\overline{s},$ and a unique limit inferior$,$ $\\underline{s}$, in the extended reals$,$ and \\begin{equation}\\label{eq:4.1.21} \\underline{s}\\le \\overline{s}. \\end{equation}"}
{"_id": "88", "title": "", "text": "If $\\{s_n\\}$ is a sequence of real numbers, then \\begin{equation}\\label{eq:4.1.22} \\lim_{n\\to\\infty} s_n=s \\end{equation} if and only if \\begin{equation}\\label{eq:4.1.23} \\limsup_{n\\to\\infty}s_n=\\liminf_{n\\to\\infty} s_n=s. \\end{equation}"}
{"_id": "89", "title": "", "text": "A sequence $\\{s_n\\}$ of real numbers converges if and only if$,$ for every  $\\epsilon>0,$ there is an integer $N$ such that \\begin{equation}\\label{eq:4.1.24} |s_n-s_m|<\\epsilon\\mbox{\\quad if\\quad} m,n\\ge N. \\end{equation}"}
{"_id": "90", "title": "", "text": "If \\begin{equation}\\label{eq:4.2.1} \\lim_{n\\to\\infty}s_n=s\\quad (-\\infty\\le s\\le\\infty), \\end{equation} then \\begin{equation}\\label{eq:4.2.2} \\lim_{k\\to\\infty} s_{n_k}=s \\end{equation} for every subsequence $\\{s_{n_k}\\}$ of $\\{s_n\\}.$"}
{"_id": "91", "title": "", "text": "If $\\{s_n\\}$ is monotonic and has a subsequence $\\{s_{n_k}\\}$ such that $$ \\lim_{k\\to\\infty} s_{n_k}=s\\quad (-\\infty\\le s\\le\\infty), $$ then $$ \\lim_{n\\to\\infty} s_n=s. $$"}
{"_id": "92", "title": "", "text": "A point $\\overline{x}$ is a limit point of a set $S$ if and only if there is a sequence $\\{x_n\\}$ of points in $S$ such that $x_n\\ne\\overline{x}$ for $n\\ge 1,$ and $$ \\lim_{n\\to\\infty}x_n=\\overline{x}. $$"}
{"_id": "93", "title": "", "text": "\\vspace*{3pt} \\begin{alist} \\item % (a)   If $\\{x_n\\}$ is bounded$,$ then $\\{x_n\\}$ has a convergent subsequence$.$ \\vspace*{3pt} \\item % (b)   If $\\{x_n\\}$ is unbounded above$,$  then $\\{x_n\\}$ has a subsequence $\\{x_{n_k}\\}$ such that $$ \\lim_{k\\to\\infty} x_{n_k}=\\infty. $$ \\vspace*{3pt} \\item % (c)   If $\\{x_n\\}$ is unbounded below$,$ then $\\{x_n\\}$ has a subsequence $\\{x_{n_k}\\}$ such that $$ \\lim_{k\\to\\infty} x_{n_k}=-\\infty. $$ \\end{alist}"}
{"_id": "94", "title": "", "text": "Let $f$ be defined on a closed interval $[a,b]$ containing $\\overline{x}.$ Then $f$ is continuous at $\\overline{x}$ $($from the right if $\\overline{x}=a,$ from the left if $\\overline{x}=b$$)$ if and only if \\begin{equation}\\label{eq:4.2.6} \\lim_{n\\to\\infty} f(x_n)=f(\\overline{x}) \\end{equation} whenever $\\{x_n\\}$ is a sequence of points in $[a,b]$ such that \\begin{equation}\\label{eq:4.2.7} \\lim_{n\\to\\infty} x_n=\\overline{x}. \\end{equation}"}
{"_id": "95", "title": "", "text": "If $f$ is continuous on a closed interval $[a,b],$ then $f$ is bounded on $[a,b].$"}
{"_id": "96", "title": "", "text": "The sum of a convergent series is unique$.$"}
{"_id": "97", "title": "", "text": "Let $$ \\sum_{n=k}^\\infty a_n=A\\mbox{\\quad and\\quad}\\sum_{n=k}^\\infty b_n=B, $$ where $A$ and $B$ are finite$.$  Then $$ \\sum_{n=k}^\\infty (ca_n)=cA $$ if $c$ is a constant$,$ $$ \\sum_{n=k}^\\infty (a_n+b_n)=A+B, $$ and $$ \\sum_{n=k}^\\infty (a_n-b_n)=A-B. $$ These relations also hold if one or both of $A$ and $B$ is infinite, provided that the right sides are not indeterminate$.$"}
{"_id": "98", "title": "Cauchy's Convergence Criterion for Series", "text": "A series $\\sum a_n$ converges if and only if for every $\\epsilon>0$ there is an integer $N$ such that \\begin{equation}\\label{eq:4.3.3} |a_n+a_{n+1}+\\cdots+a_m|<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N. \\end{equation}"}
{"_id": "99", "title": "", "text": "If $a_n\\ge0$ for $n\\ge k,$ then $\\sum a_n$ converges if its partial sums are bounded$,$ or diverges to $\\infty$ if they are not$.$ These are the only possibilities and$,$ in either case$,$ $$ \\sum_{n=k}^\\infty a_n =\\,\\sup\\set{A_n}{n\\ge k}\\negthickspace, $$ where $$ A_n=a_k+a_{k+1}+\\cdots+a_n,\\quad n\\ge k. $$"}
{"_id": "100", "title": "The Comparison Test", "text": "Suppose that \\begin{equation}\\label{eq:4.3.5} 0\\le a_n\\le b_n,\\quad n\\ge k. \\end{equation} Then \\begin{alist} \\item % (a)  $\\sum a_n<\\infty$ if  $\\sum b_n<\\infty$$.$ \\item % (b)  $\\sum b_n=\\infty$  if  $\\sum a_n=\\infty.$ \\end{alist}"}
{"_id": "101", "title": "The Integral Test", "text": "Let \\begin{equation}\\label{eq:4.3.7} c_n=f(n),\\quad n\\ge k, \\end{equation} where $f$ is  positive$,$ nonincreasing$,$ and locally integrable on $[k,\\infty).$ Then \\begin{equation}\\label{eq:4.3.8} \\sum c_n<\\infty \\end{equation} if and only if \\begin{equation}\\label{eq:4.3.9} \\int^\\infty_k f(x)\\,dx<\\infty. \\end{equation}"}
{"_id": "102", "title": "", "text": "Suppose that  $a_n\\ge0$ and $b_n>0$ for $n\\ge k.$ Then \\begin{alist} \\item % (a) $\\dst{\\sum a_n<\\infty\\mbox{\\quad if\\quad}\\sum b_n< \\infty\\mbox{\\quad and\\quad}\\limsup_{n\\to\\infty} a_n/b_n<\\infty}.$ \\item % (b)  $\\dst{\\sum a_n=\\infty\\mbox{\\quad if\\quad}\\sum b_n= \\infty\\mbox{\\quad and\\quad}\\liminf_{n\\to\\infty} a_n/b_n>0}.$ \\end{alist}"}
{"_id": "103", "title": "", "text": "Suppose that  $a_n>0,$ $b_n>0,$ and \\begin{equation}\\label{eq:4.3.12} \\frac{a_{n+1}}{ a_n}\\le \\frac{b_{n+1}}{ b_n}. \\end{equation} Then \\begin{alist} \\item % (a)  $\\sum a_n<\\infty$ if  $\\sum b_n<\\infty.$ \\item % (b)  $\\sum b_n=\\infty$  if  $\\sum a_n=\\infty.$ \\end{alist}"}
{"_id": "104", "title": "The Ratio Test", "text": "Suppose that  $a_n>0$ for $n\\ge k.$  Then \\vspace*{5pt} \\begin{alist} \\vspace*{5pt} \\item % (a) $\\sum a_n<\\infty$ if\\, $\\limsup_{n\\to\\infty} a_{n+1}/a_n<1.$ \\vspace*{5pt} \\item % (b)  $\\sum a_n=\\infty$  if\\, $\\liminf_{n\\to\\infty} a_{n+1}/a_n>1.$ \\end{alist} \\vspace*{5pt} \\noindent If \\begin{equation}\\label{eq:4.3.13} \\liminf_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}\\le1\\le \\limsup_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}, \\end{equation} then the test is inconclusive$;$ that is$,$ $\\sum a_n$ may converge or diverge$.$"}
{"_id": "105", "title": "", "text": "Suppose that  $a_n>0$ for large $n.$  Let $$ M=\\limsup_{n\\to\\infty} n\\left(\\frac{a_{n+1}}{ a_n}- 1\\right)\\mbox{\\quad and\\quad} m=\\liminf_{n\\to\\infty} n \\left(\\frac{a_{n+1}}{ a_n}-1\\right). $$ Then \\begin{alist} \\item % (a)  $\\sum a_n<\\infty$ if $M<-1.$ \\item % (b)  $\\sum a_n=\\infty$  if $m>-1.$ \\end{alist} The test is inconclusive if $m\\le-1\\le M.$"}
{"_id": "106", "title": "Cauchy's Root Test", "text": "If $a_n\\ge 0$ for $n\\ge k,$ then \\begin{alist} \\item % (a)  $\\sum a_n<\\infty$ if $\\limsup_{n\\to\\infty} a^{1/n}_n<1.$ \\item % (b)  $\\sum a_n=\\infty$ if $\\limsup_{n\\to\\infty} a^{1/n}_n>1.$ \\end{alist} The test is inconclusive if $\\limsup_{n\\to\\infty} a^{1/n}_n= 1.$"}
{"_id": "107", "title": "", "text": "absolutely$,$ then $\\sum a_n$ converges$.$"}
{"_id": "108", "title": "Dirichlet's Test for Series", "text": "The series $\\sum ^\\infty_{n=k} a_nb_n$ converges if $\\lim_{n\\to\\infty} a_n= 0,$ \\begin{equation}\\label{eq:4.3.18} \\sum |a_{n+1}-a_n|<\\infty, \\end{equation} and \\begin{equation}\\label{eq:4.3.19} |b_k+b_{k+1}+\\cdots+b_n|\\le M,\\quad n\\ge k, \\end{equation} for some constant $M.$"}
{"_id": "109", "title": "", "text": "Suppose that  $\\sum_{n=k}^\\infty a_n=A,$ where $-\\infty \\le A\\le\\infty.$ Let $\\{n_j\\}_1^\\infty$ be an increasing sequence of integers, with $n_1\\ge k$. Define \\begin{eqnarray*} b_1\\ar=a_k+\\cdots+a_{n_1},\\\\ b_2\\ar=a_{{n_1}+1}+\\cdots+a_{n_2},\\\\ &\\vdots\\\\ b_r\\ar=a_{n_{r-1}+1}+\\cdots+a_{n_r}. \\end{eqnarray*} Then $$ \\sum_{j=1}^\\infty b_{n_j}=A. $$"}
{"_id": "110", "title": "", "text": "If $\\sum_{n=1}^\\infty b_n$ is a rearrangement of an absolutely convergent series $\\sum_{n=1}^\\infty a_n,$ then $\\sum_{n=1}^\\infty b_n$ also converges absolutely$,$ and to the same sum$.$"}
{"_id": "111", "title": "", "text": "If $P=\\{a_{n_i}\\}_1^\\infty$ and $Q= \\{a_{m_j}\\}_1^\\infty$ are respectively the subsequences of all positive and negative terms in a conditionally convergent series $\\sum a_n,$ then \\begin{equation} \\label{eq:4.3.24} \\sum_{i=1}^\\infty a_{n_i}=\\infty\\mbox{\\quad and\\quad}\\sum_{j=1}^\\infty a_{m_j}=-\\infty. \\end{equation}"}
{"_id": "112", "title": "", "text": "Suppose that $\\sum_{n=1}^\\infty a_n$ is conditionally convergent and  $\\mu$ and $\\nu$ are arbitrarily given in  the extended reals$,$ with $\\mu\\le\\nu.$ Then the terms of $\\sum_{n=1}^\\infty a_n$ can be rearranged  to form a series $\\sum_{n=1}^\\infty b_n$ with partial sums $$ B_n=b_1+b_2+\\cdots+b_n,\\quad n\\ge1, $$ such that \\begin{equation}\\label{eq:4.3.25} \\limsup_{n\\to\\infty}B_n=\\nu\\mbox{\\quad and\\quad} \\liminf_{n\\to\\infty}B_n=\\mu. \\end{equation}"}
{"_id": "113", "title": "", "text": "Let $$ \\sum_{n=0}^\\infty a_n=A\\mbox{\\quad and\\quad}\\sum_{n=0}^\\infty b_n=B, $$ where $A$ and $B$ are finite, and at least one term of each series is nonzero. Then $\\sum_{n=0}^\\infty p_n=AB$ for every sequence $\\{p_n\\}$ obtained by ordering the products in $\\eqref{eq:4.3.33}$ if and only if $\\sum a_n$ and $\\sum b_n$ converge absolutely$.$ Moreover$,$ in this case, $\\sum p_n$ converges absolutely$.$"}
{"_id": "114", "title": "", "text": "If $\\sum_{n=0}^\\infty a_n$ and $\\sum_{n=0}^\\infty b_n$ converge absolutely to sums $A$ and $B,$ then the Cauchy product of $\\sum_{n=0}^\\infty a_n$ and $\\sum_{n=0}^\\infty b_n$ converges absolutely to $AB.$"}
{"_id": "115", "title": "", "text": "Let $\\{F_n\\}$ be defined on $S.$ Then \\begin{alist} \\item % (a) $\\{F_n\\}$ converges pointwise to $F$ on $S$ if and only if there is, for each $\\epsilon>0$ and $x\\in S$, an integer $N$ $($which may depend on $x$ as well as $\\epsilon)$ such that $$ |F_n(x)-F(x)|<\\epsilon\\mbox{\\quad if\\quad}\\ n\\ge N. $$ \\item % (b)  $\\{F_n\\}$ converges uniformly to $F$ on $S$ if and only if there is for each $\\epsilon>0$ an integer $N$ $($which depends only on $\\epsilon$ and not on any particular $x$ in $S)$ such that $$ |F_n(x)-F(x)|<\\epsilon\\mbox{\\quad for all $x$ in $S$ if $n\\ge N$}. $$ \\end{alist}"}
{"_id": "116", "title": "", "text": "If $\\{F_n\\}$ converges uniformly to $F$ on $S,$ then $\\{F_n\\}$ converges pointwise to $F$ on $S.$ The converse is false$;$ that is$,$ pointwise convergence does not imply uniform convergence."}
{"_id": "117", "title": "Cauchy's Uniform Convergence Criterion", "text": "A sequence of functions $\\{F_n\\}$ converges uniformly on a set $S$ if and only if for each $\\epsilon>0$ there is an integer $N$ such that \\begin{equation} \\label{eq:4.4.2} \\|F_n-F_m\\|_S<\\epsilon\\mbox{\\quad if\\quad} n, m\\ge N. \\end{equation}"}
{"_id": "118", "title": "", "text": "If $\\{F_n\\}$ converges uniformly to $F$ on $S$ and each $F_n$ is continuous at a point $x_0$ in $S,$ then so is $F$. Similar statements hold for continuity from the right and left$.$"}
{"_id": "119", "title": "", "text": "Suppose that  $\\{F_n\\}$ converges uniformly to $F$ on $S=[a,b]$. Assume that $F$ and all $F_n$ are   integrable on $[a,b].$ Then \\begin{equation} \\label{eq:4.4.10} \\int_a^b F(x)\\,dx=\\lim_{n\\to\\infty}\\int_a^b F_n(x)\\,dx. \\end{equation}"}
{"_id": "120", "title": "", "text": "Suppose that  $\\{F_n\\}$ converges pointwise to $F$ and each $F_n$ is integrable on $[a,b].$ \\begin{alist} \\item % (a) If the convergence is uniform$,$ then $F$ is integrable on $[a,b]$ and $\\eqref{eq:4.4.10}$ holds. \\item % (b) If the sequence $\\{\\|F_n\\|_{[a,b]}\\}$ is bounded and $F$ is integrable on $[a,b],$ then $\\eqref{eq:4.4.10}$ holds. \\end{alist}"}
{"_id": "121", "title": "", "text": "Suppose that  $F'_n$ is continuous on $[a,b]$ for all $n\\ge1$ and $\\{F'_n\\}$ converges uniformly on $[a,b].$ Suppose also that  $\\{F_n(x_0)\\}$ converges for some $x_0$ in $[a,b].$ Then $\\{F_n\\}$ converges uniformly on $[a,b]$ to a differentiable limit function $F,$ and \\begin{equation} \\label{eq:4.4.11} F'(x)=\\lim_{n\\to\\infty}F'_n(x),\\quad a<x<b, \\end{equation} while \\begin{equation} \\label{eq:4.4.12} F'_+(a)=\\lim_{n\\to\\infty}F'_n(a+)\\mbox{\\quad and\\quad} F'_-(b)= \\lim_{n\\to\\infty}F'_n(b-). \\end{equation}"}
{"_id": "122", "title": "Cauchy's  Uniform Convergence Criterion", "text": "A series $\\sum f_n$ converges uniformly on a set $S$ if and only if for each $\\epsilon>0$ there is an integer $N$ such that \\vskip0pt \\begin{equation} \\label{eq:4.4.16} \\|f_n+f_{n+1}+\\cdots+f_m\\|_S<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N. \\end{equation}"}
{"_id": "123", "title": "Weierstrass's Test", "text": "The series $\\sum f_n$ converges uniformly on $S$ if \\begin{equation} \\label{eq:4.4.17} \\|f_n\\|_S\\le M_n,\\quad n\\ge k, \\end{equation} where $\\sum M_n<\\infty.$"}
{"_id": "124", "title": "Dirichlet's Test for Uniform Convergence", "text": "The series $$ \\sum_{n=k}^\\infty  f_ng_n $$  converges uniformly on $S$ if  $\\{f_n\\}$ converges uniformly to zero on $S,$  $\\sum (f_{n+1}-f_n)$ converges absolutely uniformly on $S,$ and \\begin{equation} \\label{eq:4.4.19} \\|g_k+g_{k+1}+\\cdots+g_n\\|_S\\le M,\\quad n\\ge k, \\end{equation} for some constant $M.$"}
{"_id": "125", "title": "", "text": "If $\\sum_{n=k}^\\infty  f_n$ converges uniformly to $F$ on $S$ and each $f_n$ is continuous at a point $x_0$ in $S,$ then so is $F.$ Similar statements hold for continuity from the right and left$.$"}
{"_id": "126", "title": "", "text": "Suppose that  $\\sum_{n=k}^\\infty f_n$ converges uniformly to $F$ on $S=[a,b].$ Assume that $F$ and  $f_n,$ $n\\ge k,$ are   integrable on $[a,b].$ Then $$ \\int_a^b F(x)\\,dx=\\sum_{n=k}^\\infty \\int_a^b f_n(x)\\,dx. $$"}
{"_id": "127", "title": "", "text": "Suppose that  $f_n$ is continuously differentiable on $[a,b]$ for each $n\\ge k,$ $\\sum_{n=k}^\\infty f_n(x_0)$ converges for some $x_0$ in $[a,b],$ and $\\sum_{n=k}^\\infty  f'_n$ converges uniformly on $[a,b].$ Then $\\sum_{n=k}^\\infty f_n$ converges uniformly on $[a,b]$ to a differentiable function $F,$ and $$ F'(x)=\\sum_{n=k}^\\infty  f'_n(x),\\quad a<x<b, $$ while $$ F'(a+)=\\sum_{n=k}^\\infty  f'_n(a+)\\mbox{\\quad and\\quad} F'(b-)= \\sum_{n=k}^\\infty  f'_n(b-). $$"}
{"_id": "128", "title": "", "text": "In connection with the power series $\\eqref{eq:4.5.1},$ define $R$ in the extended reals by \\begin{equation}\\label{eq:4.5.2} \\frac{1}{ R}=\\limsup_{n\\to\\infty} |a_n|^{1/n}. \\end{equation} In particular$,$ $R=0$ if $\\limsup_{n\\to\\infty} |a_n|^{1/n}= \\infty$, and $R=\\infty$ if $\\limsup_{n\\to\\infty} |a_n|^{1/n}=0.$ Then the power series converges \\begin{alist} \\item % (a)  only for $x=x_0$ if $R=0;$ \\item % (b)  for all $x$ if $R=\\infty,$ and absolutely uniformly in every bounded set$;$ \\item % (c) for $x$ in $(x_0-R, x_0+R)$ if $0<R<\\infty,$ and absolutely uniformly in every closed subset of this interval. \\end{alist} The series diverges if $|x-x_0|>R.$ No general statement can be made concerning convergence at the endpoints $x=x_0+R$ and $x=x_0-R:$ the series may converge absolutely or conditionally at both$,$ converge conditionally at one and diverge at the other$,$ or diverge at both$.$"}
{"_id": "129", "title": "", "text": "The radius of convergence of $\\sum a_n(x-x_0)^n$ is given by $$ \\frac{1}{ R}=\\lim_{n\\to\\infty}\\left|\\frac{a_{n+1}}{a_n}\\right| $$ if the limit exists in the extended reals$.$"}
{"_id": "130", "title": "", "text": "A power series $$ f(x)=\\sum^\\infty_{n=0} a_n(x-x_0)^n $$ \\newpage \\noindent with positive radius of convergence $R$ is continuous and differentiable in its interval of convergence$,$ and its derivative can be obtained by differentiating term by term$;$ that is$,$ \\begin{equation} \\label{eq:4.5.9} f'(x)=\\sum^\\infty_{n=1} na_n(x-x_0)^{n-1}, \\end{equation} which can also be written as \\begin{equation} \\label{eq:4.5.10} f'(x)=\\sum^\\infty_{n=0}(n+1)a_{n+1} (x-x_0)^n. \\end{equation} This series also has radius of convergence $R.$"}
{"_id": "131", "title": "", "text": "A power series $$ f(x)=\\sum_{n=0}^\\infty a_n(x-x_0)^n $$ with positive radius of convergence $R$  has derivatives of all orders in its interval of convergence$,$ which can be obtained by repeated term by term differentiation$;$ thus$,$ \\begin{equation}\\label{eq:4.5.11} f^{(k)}(x)=\\sum^\\infty_{n=k} n(n-1)\\cdots (n-k+1)a_n(x-x_0)^{n-k}. \\end{equation} The radius of convergence of each of these series is $R.$"}
{"_id": "132", "title": "", "text": "If $x_1$ and $x_2$ are in the interval of convergence of $$ f(x)=\\sum^\\infty_{n=0} a_n(x-x_0)^n, $$ then $$ \\int_{x_1}^{x_2} f(x)\\,dx=\\sum^\\infty_{n=0}\\frac{a_n}{ n+1}\\left[(x_2-x_0)^{n+1}- (x_1-x_0)^{n+1}\\right]; $$ that is$,$ a power series may be integrated term by term between any two points in its interval of convergence$.$"}
{"_id": "133", "title": "", "text": "Suppose that  $f$ is infinitely differentiable on an interval $I$ and \\begin{equation}\\label{eq:4.5.18} \\lim_{n\\to\\infty}\\frac{r^n}{ n!}\\|f^{(n)}\\|_I=0. \\end{equation} Then$,$ if $x_0\\in I^0,$ the Taylor series $$ \\sum^\\infty_{n=0}\\frac{f^{(n)}(x_0)}{ n!} (x-x_0)^n $$  converges uniformly to $f$ on $$ I_r=I\\cap [x_0-r,x_0+r]. $$"}
{"_id": "134", "title": "", "text": "If \\begin{equation}\\label{eq:4.5.22} f(x)=\\sum^\\infty_{n=0} a_n(x-x_0)^n,\\quad |x-x_0|<R_1, \\end{equation} \\begin{equation}\\label{eq:4.5.23} g(x)=\\sum^\\infty_{n=0} b_n(x-x_0)^n,\\quad |x-x_0|<R_2, \\end{equation} and $\\alpha$ and $\\beta$ are constants$,$ then $$ \\alpha f(x)+\\beta g(x)=\\sum^\\infty_{n=0} (\\alpha a_n+\\beta b_n)(x-x_0)^n,\\quad |x-x_0|<R, $$ where  $R\\ge\\min\\{R_1,R_2\\}.$"}
{"_id": "135", "title": "", "text": "If $f$ and $g$ are given by $\\eqref{eq:4.5.22}$ and $\\eqref{eq:4.5.23},$ then \\begin{eqnarray} f(x)g(x)\\ar=\\sum^\\infty_{n=0} c_n(x-x_0)^n,\\quad|x-x_0|<R,\\label{eq:4.5.24}\\\\ \\arraytext{where}\\nonumber c_n\\ar=\\sum^n_{r=0} a_rb_{n-r}=\\sum^n_{r=0} a_{n-r}b_r\\nonumber \\end{eqnarray} and $R\\ge\\min\\{R_1,R_2\\}.$"}
{"_id": "136", "title": "Abel's Theorem", "text": "Let $f$ be defined by a power series $\\eqref{eq:4.5.29}$ with finite radius of convergence $R.$ \\begin{alist} \\item % (a)  If $\\sum^\\infty_{n=0} a_nR^n$ converges$,$ then $$ \\lim_{x\\to (x_0+R)-}f(x)=\\sum^\\infty_{n=0} a_nR^n. $$ \\item % (b)  If $\\sum^\\infty_{n=0} (-1)^na_nR^n$ converges$,$ then $$ \\lim_{x\\to (x_0-R)+} f(x)=\\sum^\\infty_{n=0} (-1)^na_nR^n. $$ \\end{alist}"}
{"_id": "137", "title": "", "text": "If $\\mathbf{X},$ $\\mathbf{Y},$ and $\\mathbf{Z}$ are in $\\R^n$ and $a$ and $b$ are real numbers$,$ then \\begin{alist} \\item % (a)  $\\mathbf{X}+\\mathbf{Y}=\\mathbf{Y}+\\mathbf{X}$  $($vector addition is commutative$).$ \\item % (b) $(\\mathbf{X}+\\mathbf{Y})+\\mathbf{Z}=\\mathbf{X}+(\\mathbf{Y}+\\mathbf{Z})$ $($vector addition is associative$).$ \\item % (c)  There is a unique vector $\\mathbf{0},$ called the zero vector$,$ such that $\\mathbf{X}+\\mathbf{0}=\\mathbf{X}$ for all $\\mathbf{X}$ in $\\R^n.$ \\item % (d)  For each $\\mathbf{X}$ in $\\R^n$ there is a unique vector $-\\mathbf{X}$ such that $\\mathbf{X}+(-\\mathbf{X})=\\mathbf{0}.$ \\item % (e)  $a(b\\mathbf{X})=(ab)\\mathbf{X}.$ \\item % (f) $(a+b)\\mathbf{X}=a\\mathbf{X}+b\\mathbf{X}.$ \\item % (g)  $a(\\mathbf{X}+\\mathbf{Y})=a\\mathbf{X}+a\\mathbf{Y}.$ \\item % (h)  $1\\mathbf{X}=\\mathbf{X}.$ \\end{alist}"}
{"_id": "138", "title": "Triangle Inequality", "text": "If $\\mathbf{X}$ and $\\mathbf{Y}$ are in $\\R^n,$ then \\begin{equation}\\label{eq:5.1.6} |\\mathbf{X}+\\mathbf{Y}|\\le |\\mathbf{X}|+|\\mathbf{Y}|, \\end{equation} with equality if and only if one of the vectors is a nonnegative multiple of the other$.$"}
{"_id": "139", "title": "", "text": "If $\\mathbf{X},$ $\\mathbf{Y},$ and $\\mathbf{Z}$ are members of $\\R^n$ and $a$ is a scalar, then \\begin{alist} \\item % (a)  $|a\\mathbf{X}|=|a|\\,|\\mathbf{X}|.$ \\item % (b)  $|\\mathbf{X}|\\ge0,$ with equality if and only if $\\mathbf{X}= \\mathbf{0}.$ \\item % (c)  $|\\mathbf{X}-\\mathbf{Y}|\\ge0,$ with equality if and only if $\\mathbf{X}=\\mathbf{Y}.$ \\item % (d) $\\mathbf{X}\\cdot\\mathbf{Y}=\\mathbf{Y}\\cdot\\mathbf{X}.$ \\item % (e)  $\\mathbf{X}\\cdot (\\mathbf{Y}+\\mathbf{Z})=\\mathbf{X}\\cdot\\mathbf{Y}+ \\mathbf{X}\\cdot\\mathbf{Z}.$ \\item % (f)  $(c\\mathbf{X})\\cdot\\mathbf{Y}=\\mathbf{X}\\cdot (c\\mathbf{Y})= c(\\mathbf{X}\\cdot\\mathbf{Y}).$ \\end{alist}"}
{"_id": "140", "title": "", "text": "Let $$ \\overline{\\mathbf{X}}=(\\overline{x}_1,\\overline{x}_2, \\dots,\\overline{x}_n) \\mbox{\\quad and\\quad}\\mathbf{X}_r=(x_{1r}, x_{2r}, \\dots, x_{nr}),\\quad r\\ge1. $$ Then $\\lim_{r\\to\\infty}\\mathbf{X}_r=\\overline{\\mathbf{X}}$ if and only if $$ \\lim_{r\\to\\infty}x_{ir}=\\overline{x}_i,\\quad 1\\le i\\le n; $$ that is$,$ a sequence $\\{\\mathbf{X}_r\\}$ of points in $\\R^n$ converges to a limit $\\overline{\\mathbf{X}}$ if and only if the sequences of components of $\\{\\mathbf{X}_r\\}$ converge to the respective components of $\\overline{\\mathbf{X}}.$"}
{"_id": "141", "title": "Cauchy's Convergence Criterion", "text": "A sequence $\\{\\mathbf{X}_r\\}$  in $\\R^n$ converges if and only if for each $\\epsilon>0$ there is an integer $K$ such that $$ |\\mathbf{X}_r-\\mathbf{X}_s|<\\epsilon\\mbox{\\quad if\\quad} r,s\\ge K. $$"}
{"_id": "142", "title": "Principle of Nested Sets", "text": "If $S_1,$ $S_2,$ \\dots\\ are closed nonempty subsets of $\\R^n$ such that \\begin{equation}\\label{eq:5.1.14} S_1\\supset S_2\\supset\\cdots\\supset S_r\\supset\\cdots \\end{equation} and \\begin{equation}\\label{eq:5.1.15} \\lim_{r\\to\\infty} d(S_r)=0, \\end{equation} then the intersection $$ I=\\bigcap^\\infty_{r=1}S_r $$ contains exactly one point$.$"}
{"_id": "143", "title": "Heine--Borel Theorem", "text": "If ${\\mathcal H}$ is an open covering of a compact subset $S,$  then $S$ can be covered by finitely many sets from ${\\mathcal H}.$"}
{"_id": "144", "title": "", "text": "An open set $S$ in $\\R^n$ is connected if and only if it is polygonally connected$.$"}
{"_id": "145", "title": "", "text": "If $\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})$ exists$,$ then it is unique."}
{"_id": "146", "title": "", "text": "Suppose that  $f$ and $g$ are defined on a set $D,$ $\\mathbf{X}_0$ is a limit point of $D,$ and $$ \\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=L_1,\\quad\\lim_{\\mathbf{X}\\to\\mathbf{X}_0} g(\\mathbf{X})=L_2. $$ Then \\begin{eqnarray} \\lim_{\\mathbf{X}\\to\\mathbf{X}_0}(f+g)(\\mathbf{X})\\ar=L_1+L_2,\\label{eq:5.2.10}\\\\ \\lim_{\\mathbf{X}\\to\\mathbf{X}_0}(f-g)(\\mathbf{X})\\ar=L_1-L_2,\\label{eq:5.2.11}\\\\ \\lim_{\\mathbf{X}\\to\\mathbf{X}_0}(fg)(\\mathbf{X})\\ar=L_1L_2,\\label{eq:5.2.12}\\\\ \\arraytext{and$,$ if $L_2\\ne0,$}\\nonumber\\\\ \\lim_{\\mathbf{X}\\to\\mathbf{X}_0}\\left(\\frac{f}{ g}\\right)(\\mathbf{X}) \\ar=\\frac{L_1}{ L_2}.\\label{eq:5.2.13} \\end{eqnarray}"}
{"_id": "147", "title": "", "text": "Suppose that  $\\mathbf{X}_0$ is in $D_f$ and is a limit point of $D_f.$ Then $f$ is continuous at $\\mathbf{X}_0$ if and only if for each $\\epsilon>0$ there is a $\\delta>0$ such that $$ \\left|f(\\mathbf{X})-f(\\mathbf{X}_0)\\right|<\\epsilon $$ whenever $$ |\\mathbf{X}-\\mathbf{X}_0|<\\delta\\mbox{\\quad and\\quad}\\mathbf{X}\\in D_f. $$"}
{"_id": "148", "title": "", "text": "If $f$ and $g$ are continuous on a set $S$ in $\\R^n,$ then so are $f+g,$ $f-g,$ and $fg.$ Also$,$ $f/g$ is continuous at each $\\mathbf{X}_0$ in $S$ such that $g(\\mathbf{X}_0)\\ne0.$"}
{"_id": "149", "title": "", "text": "For a vector-valued function $\\mathbf{G},$ $$ \\lim_{\\mathbf{U}\\to\\mathbf{U}_0}\\mathbf{G}(\\mathbf{U})=\\mathbf{L} $$ if and only if for each $\\epsilon>0$ there is a $\\delta>0$ such that $$ |\\mathbf{G}(\\mathbf{U})-\\mathbf{L}|<\\epsilon\\mbox{\\quad whenever\\quad} 0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta\\mbox{\\quad and\\quad}\\mathbf{U}\\in D_{\\mathbf{G}}. $$ Similarly, $\\mathbf{G}$ is continuous at $\\mathbf{U}_0$ if and only if for each $\\epsilon> 0$ there is a $\\delta>0$ such that $$ |\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|<\\epsilon \\mbox{\\quad whenever\\quad}  |\\mathbf{U}-\\mathbf{U}_0|<\\delta\\mbox{\\quad and\\quad}\\mathbf{U}\\in D_{\\mathbf{G}}. $$"}
{"_id": "150", "title": "", "text": "Let $f$ be a real-valued function defined on a subset of $\\R^n,$  and let the vector-valued function $\\mathbf{G}=(g_1,g_2, \\dots,g_n)$ be defined on a domain $D_\\mathbf{G}$ in $\\R^m.$ Let the set $$ T=\\set{\\mathbf{U}}{\\mathbf{U}\\in D_{\\mathbf{G}}\\mbox{\\quad and \\quad} \\mathbf{G}(\\mathbf{U})\\in D_f} $$ $($Figure~\\ref{figure:5.2.3}$)$,  be nonempty$,$ and define the real-valued composite function $$ h=f\\circ\\mathbf{G} $$ on $T$ by $$ h(\\mathbf{U})=f(\\mathbf{G}(\\mathbf{U})),\\quad \\mathbf{U}\\in T. $$ Now suppose that  $\\mathbf{U}_0$ is in $T$ and is a limit point of $T,$ $\\mathbf{G}$ is continuous at $\\mathbf{U}_0,$ and $f$ is continuous at $\\mathbf{X}_0=\\mathbf{G}(\\mathbf{U}_0).$ Then $h$ is continuous at $\\mathbf{U}_0.$"}
{"_id": "151", "title": "", "text": "If $f$ is continuous on a compact set $S$ in $\\R^n,$ then $f$ is bounded on~$S.$"}
{"_id": "152", "title": "", "text": "Let $f$ be continuous on a compact set $S$ in $\\R^n$ and $$ \\alpha=\\inf_{\\mathbf{X}\\in S}f(\\mathbf{X}),\\quad\\beta= \\sup_{\\mathbf{X}\\in S}f(\\mathbf{X}). $$ Then $$ f(\\mathbf{X}_1)=\\alpha\\mbox{\\quad and\\quad} f(\\mathbf{X}_2)=\\beta $$ for some $\\mathbf{X}_1$ and $\\mathbf{X}_2$ in $S.$"}
{"_id": "153", "title": "Intermediate Value Theorem", "text": "Let $f$ be continuous on a region $S$ in $\\R^n.$ Suppose that $\\mathbf{A}$ and $\\mathbf{B}$ are in $S$ and $$ f(\\mathbf{A})<u<f(\\mathbf{B}). $$ Then $f(\\mathbf{C})=u$ for some $\\mathbf{C}$ in $S.$"}
{"_id": "154", "title": "", "text": "If $f$ is continuous on a compact set $S$ in $\\R^n,$ then $f$ is uniformly continuous on $S.$"}
{"_id": "155", "title": "", "text": "If $f_{x_i} (\\mathbf{X})$ and $g_{x_i} (\\mathbf{X})$ exist$,$ then \\begin{eqnarray*} \\frac{\\partial (f+g)(\\mathbf{X})}{\\partial x_i}\\ar=f_{x_i}(\\mathbf{X})+ g_{x_i}(\\mathbf{X}),\\\\ \\frac{\\partial (fg)(\\mathbf{X})}{\\partial x_i}\\ar=f_{x_i}(\\mathbf{X}) g(\\mathbf{X})+f(\\mathbf{X})g_{x_i} (\\mathbf{X}), \\end{eqnarray*} \\newpage \\noindent and$,$ if $g(\\mathbf{X})\\ne0,$ $$ \\frac{\\partial (f/g)(\\mathbf{X})}{\\partial x_i}=\\frac{g(\\mathbf{X})f_{x_i} (\\mathbf{X})- f(\\mathbf{X})g_{x_i}(\\mathbf{X})}{[g(\\mathbf{X})]^2}. $$"}
{"_id": "156", "title": "", "text": "Suppose that  $f,$ $f_x,$ $f_y,$ and $f_{xy}$ exist on a neighborhood $N$ of $(x_0,y_0),$ and $f_{xy}$ is continuous at $(x_0,y_0).$ Then $f_{yx}(x_0,y_0)$ exists, and \\begin{equation}\\label{eq:5.3.5} f_{yx}(x_0,y_0)=f_{xy}(x_0,y_0). \\end{equation}"}
{"_id": "157", "title": "", "text": "Suppose that  $f$ and all its partial derivatives of order $\\le r$ are continuous on an open subset $S$ of $\\R^n.$  Then \\begin{equation}\\label{eq:5.3.11} f_{x_{i_1}x_{i_2}, \\dots, x_{i_r}}(\\mathbf{X})=f_{x_{j_1}x_{j_2}, \\dots, x_{j_r}}(\\mathbf{X}),\\quad \\mathbf{X}\\in S, \\end{equation} if each of the variables $x_1,$ $x_2,$ \\dots$,$ $x_n$ appears the same number of times in $$ \\{x_{i_1}, x_{i_2}, \\dots,x_{i_r}\\}\\mbox{\\quad and \\quad} \\{x_{j_1},x_{j_2}, \\dots,x_{j_r}\\}. $$  If this number is $r_k,$ we denote the common value of the two sides of $\\eqref{eq:5.3.11}$ by \\begin{equation}\\label{eq:5.3.12} \\frac{\\partial^r f(\\mathbf{X})}{\\partial x^{r_1}_1\\partial x^{r_2}_2\\cdots \\partial x^{r_n}_n}, \\end{equation} it being understood that \\begin{equation}\\label{eq:5.3.13} 0\\le r_k\\le r,\\quad 1\\le k\\le n, \\end{equation} \\begin{equation}\\label{eq:5.3.14} r_1+r_2+\\cdots+r_n=r, \\end{equation} and$,$ if $r_k=0,$ we omit the symbol $\\partial x_k^0$  from the ``denominator'' of $\\eqref{eq:5.3.12}.$"}
{"_id": "158", "title": "", "text": "If $f$ is differentiable at $\\mathbf{X}_0=(x_{10},x_{20}, \\dots,x_{n0}),$ then $f_{x_1}(\\mathbf{X}_0),$ $f_{x_2}(\\mathbf{X}_{0}),$ \\dots$,$ $f_{x_n}(\\mathbf{X}_0)$ exist and the constants  $m_1,$ $m_2,$ \\dots$,$ $m_n$ in $\\eqref{eq:5.3.16}$ are given by \\begin{equation}\\label{eq:5.3.18} m_i=f_{x_i}(\\mathbf{X}_0),\\quad 1\\le i\\le n; \\end{equation} that is$,$ $$ \\lim_{\\mathbf{X}\\to\\mathbf{X}_0} \\frac{f(\\mathbf{X})-f(\\mathbf{X}_0)- \\dst{\\sum^n_{i=1}}\\, f_{x_i}(\\mathbf{X}_0) (x_i-x_{i0})} { |\\mathbf{X}-\\mathbf{X}_0|}=0. $$"}
{"_id": "159", "title": "", "text": "If $f$ is differentiable at $\\mathbf{X}_0,$ then $f$ is continuous at $\\mathbf{X}_0$."}
{"_id": "160", "title": "", "text": "$\\mathbf{X}_0,$ then so are $f+g$ and $fg$. The same is true of $f/g$ if $g(\\mathbf{X}_0)\\ne0$. The differentials are given by \\begin{eqnarray*} d_{\\mathbf{X}_0}(f+g)\\ar=d_{\\mathbf{X}_0}f+d_{\\mathbf{X}_0}g,\\\\ d_{\\mathbf{X}_0}(fg)\\ar=f(\\mathbf{X}_0)d_{\\mathbf{X}_0} g+g(\\mathbf{X}_0) d_{\\mathbf{X}_0}f,\\\\ \\noalign{\\hbox{and}} d_{\\mathbf{X}_0}\\left(\\frac{f}{ g}\\right)\\ar=\\frac{g(\\mathbf{X}_0)d_{\\mathbf{X}_0}f-f(\\mathbf{X}_0) d_{\\mathbf{X}_0}g}{[g(\\mathbf{X}_0)]^2}. \\end{eqnarray*}"}
{"_id": "161", "title": "", "text": "If $f_{x_1},$ $f_{x_2},$ \\dots$,$ $f_{x_n}$ exist on a neighborhood of $\\mathbf{X}_0$ and are continuous at $\\mathbf{X}_0,$ then $f$ is differentiable at $\\mathbf{X}_0.$"}
{"_id": "162", "title": "", "text": "Suppose that  $f$ is defined in a neighborhood of $\\mathbf{X}_0$ in $\\R^n$  and $f_{x_1}(\\mathbf{X}_0),$ $f_{x_2}(\\mathbf{X}_{0}),$  \\dots$,$ $f_{x_n}(\\mathbf{X}_{0})$  exist$.$ Let $\\mathbf{X}_0$ be a local extreme point of $f.$ Then \\begin{equation}\\label{eq:5.3.42} f_{x_i}(\\mathbf{X}_0)=0,\\quad 1\\le i\\le n. \\end{equation}"}
{"_id": "163", "title": "The Chain Rule", "text": "Suppose that  the real-valued function $f$ is differentiable at $\\mathbf{X}_0$ in $\\R^n,$ the vector-valued function $\\mathbf{G} =(g_1,g_2, \\dots,g_n)$ is differentiable at $\\mathbf{U}_0$ in $\\R^m,$ and $\\mathbf{X}_{0}  = \\mathbf{G}(\\mathbf{U}_0).$ Then the real-valued composite function $h=f\\circ\\mathbf{G}$ defined by \\begin{equation} \\label{eq:5.4.3} h(\\mathbf{U})=f(\\mathbf{G}(\\mathbf{U})) \\end{equation} is differentiable at $\\mathbf{U}_0,$ and \\begin{equation} \\label{eq:5.4.4} d_{\\mathbf{U}_0}h=f_{x_1}(\\mathbf{X}_0) d_{\\mathbf{U}_0}g_1+f_{x_2} (\\mathbf{X}_0) d_{\\mathbf{U}_0}g_2+\\cdots +f_{x_n} (\\mathbf{X}_0) d_{\\mathbf{U}_0}g_n. \\end{equation}"}
{"_id": "164", "title": "Mean Value Theorem for Functions of $\\mathbf n$ Variables", "text": "Let $f$ be continuous at $\\mathbf{X}_1=(x_{11},x_{21}, \\dots, x_{n1})$ and $\\mathbf{X}_2=(x_{12},x_{22}, \\dots,x_{n2})$ and differentiable on the line segment $L$ from $\\mathbf{X}_1$ to $\\mathbf{X}_2.$ Then \\begin{equation} \\label{eq:5.4.21} f(\\mathbf{X}_2)-f(\\mathbf{X}_1)=\\sum_{i=1}^n f_{x_i} (\\mathbf{X}_0)(x_{i2}-x_{i1})=(d_{\\mathbf{X}_0}f)(\\mathbf{X}_2 -\\mathbf{X}_1) \\end{equation} for some $\\mathbf{X}_0$ on $L$ distinct from $\\mathbf{X}_1$ and $\\mathbf{X}_2$."}
{"_id": "165", "title": "Taylor's Theorem for Functions of $\\mathbf n$ Variables", "text": "Suppose \\\\that  $f$ and its partial derivatives of order $\\le k$ are differentiable at $\\mathbf{X}_0$ and $\\mathbf{X}$ in $\\R^n$ and on the line segment $L$ connecting them$.$ Then \\begin{equation} \\label{eq:5.4.25} f(\\mathbf{X})=\\sum_{r=0}^k\\frac{1}{ r!} (d^{(r)}_{\\mathbf{X}_0}f)  (\\mathbf{X}-\\mathbf{X})+\\frac{1}{(k+1)!} (d^{(k+1)}_{\\widetilde{\\mathbf{X}}} f)(\\mathbf{X}-{\\mathbf{X}_0}) \\end{equation} for some $\\widetilde{\\mathbf{X}}$  on $L$ distinct from $\\mathbf{X}_0$ and $\\mathbf{X}$."}
{"_id": "166", "title": "", "text": "Suppose that  $f$ and its partial derivatives of order $\\le k-1$ are differentiable in a neighborhood $N$ of a point $\\mathbf{X}_0$ in $\\R^n$ and all $k$th-order partial derivatives of $f$ are continuous at $\\mathbf{X}_0.$ Then \\begin{equation} \\label{eq:5.4.31} \\lim_{\\mathbf{X}\\to\\mathbf{X}_0} \\frac{f(\\mathbf{X})-T_k(\\mathbf{X})}{ |\\mathbf{X}-\\mathbf{X}_0|^k}=0. \\end{equation}"}
{"_id": "167", "title": "", "text": "Suppose that  $f$ satisfies the hypotheses of Theorem~$\\ref{thmtype:5.4.9}$ with $k\\ge2,$ and  \\begin{equation} \\label{eq:5.4.38} d^{(r)}_{\\mathbf{X}_0} f\\equiv0\\quad (1\\le r\\le k-1),\\quad d^{(k)}_\\mathbf{X_0} f\\not\\equiv0. \\end{equation} Then \\begin{alist} \\item % (a) $\\mathbf{X}_0$ is not a local extreme point of $f$ unless $d^{(k)}_{\\mathbf{X}_0}f$ is semidefinite as a polynomial in $\\mathbf{X}-\\mathbf{X}_0.$ In particular$,$  $\\mathbf{X}_0$ is not a local extreme point of $f$ if $k$ is odd$.$ \\item % (b)  $\\mathbf{X}_0$ is a local minimum point of $f$ if $d^{(k)}_{\\mathbf{X}_0} f$ is positive definite$,$ or a local maximum point if $d^{(k)}_{\\mathbf{X}_0}f$ is negative definite$.$ \\item % (c)  If $d^{(k)}_{\\mathbf{X}_0}f$ is semidefinite$,$ then $\\mathbf{X}_0$ may be a local extreme point of $f,$ but it need not be$.$ \\end{alist}"}
{"_id": "168", "title": "", "text": "A transformation $\\mathbf{L}: \\R^n \\to \\R^m$ defined on all of $\\R^n$ is linear if and only if \\begin{equation}\\label{eq:6.1.1} \\mathbf{L}(\\mathbf{X})=\\left[\\begin{array}{c} a_{11}x_1+a_{12}x_2+ \\cdots+a_{1n}x_n\\\\a_{21}x_1+a_{22}x_2+\\cdots+a_{2n}x_n\\\\ \\vdots\\\\a_{m1}x_1+a_{m2}x_2+\\cdots+a_{mn}x_n\\end{array}\\right], \\end{equation} where the $a_{ij}$'s are constants$.$"}
{"_id": "169", "title": "", "text": "If $\\mathbf{A},$ $\\mathbf{B},$ and $\\mathbf{C}$ are $m\\times n$ matrices$,$ then $$ (\\mathbf{A}+\\mathbf{B})+\\mathbf{C}=\\mathbf{A}+(\\mathbf{B} +\\mathbf{C}). $$"}
{"_id": "170", "title": "", "text": "If $\\mathbf{A}$ and $\\mathbf{B}$ are $m\\times n$ matrices and $r$ and $s$ are real numbers$,$ then \\part{a} $r(s\\mathbf{A}) =(rs)\\mathbf{A};$ \\part{b} $(r+s)\\mathbf{A}=r\\mathbf{A}+s\\mathbf{A};$ \\part{c} $r(\\mathbf{A}+\\mathbf{B})=r\\mathbf{A}+r\\mathbf{B}.$"}
{"_id": "171", "title": "", "text": "If $\\mathbf{A},$ $\\mathbf{B},$ and $\\mathbf{C}$ are $m\\times p,$ $p\\times q,$ and $q\\times n$ matrices$,$ respectively$,$ then $(\\mathbf{AB})\\mathbf{C}=\\mathbf{A}(\\mathbf{BC}).$"}
{"_id": "172", "title": "", "text": "\\begin{alist} \\item % (a) If we regard the vector $$ \\mathbf{X}=\\left[\\begin{array}{c} x_1\\\\ x_2\\\\\\vdots\\\\ x_n\\end{array}\\right] $$ as an $n\\times 1$ matrix$,$ then the linear transformation $\\eqref{eq:6.1.1}$ can be written as $$ \\mathbf{L}(\\mathbf{X})=\\mathbf{AX}. $$ \\newpage \\noindent \\item % (b) If $\\mathbf{L}_1$ and $\\mathbf{L}_2$ are linear transformations from $\\R^n$ to $\\R^m$ with matrices $\\mathbf{A}_1$ and $\\mathbf{A}_{2}$ respectively$,$ then $c_1\\mathbf{L}_1+c_2\\mathbf{L}_2$ is the linear transformation from $\\R^n$ to $\\R^m$ with matrix $c_1\\mathbf{A}_1+c_2\\mathbf{A}_{2}.$ \\item % (c) If $\\mathbf{L}_1: \\R^n\\to \\R^p$ and $\\mathbf{L}_2: \\R^p\\to \\R^m$ are linear transformations with matrices $\\mathbf{A}_1$ and $\\mathbf{A}_2,$ respectively$,$ then the composite function $\\mathbf{L}_3=\\mathbf{L}_2\\circ\\mathbf{L}_1,$ defined by $$ \\mathbf{L}_3(\\mathbf{X})=\\mathbf{L}_2(\\mathbf{L}_1(\\mathbf{X})), $$ is the linear transformation from $\\R^n$ to $\\R^m$ with matrix $\\mathbf{A}_2\\mathbf{A}_1.$ \\end{alist}"}
{"_id": "173", "title": "", "text": "If $\\mathbf{A}$ and $\\mathbf{B}$ are $n\\times n$  matrices$,$ then $$ \\det(\\mathbf{A}\\mathbf{B})=\\det(\\mathbf{A})\\det(\\mathbf{B}). $$"}
{"_id": "174", "title": "", "text": "Let $\\mathbf{A}$ be an $n\\times n$ matrix$.$ \\begin{alist} \\item % (a) The sum of the products of the entries of a  row of $\\mathbf{A}$ and their cofactors equals $\\det(\\mathbf{A}),$ while the  sum of the products of the entries of a  row of $\\mathbf{A}$ and the cofactors of the entries of a different row equals zero$;$ that is$,$ \\begin{equation} \\label{eq:6.1.8} \\sum^n_{k=1} a_{ik}c_{jk}=\\left\\{\\casespace\\begin{array}{ll}\\det(\\mathbf{A}),&i=j,\\\\  0,&i\\ne j.\\end{array}\\right. \\end{equation} \\item % (b) The sum of the products of the entries of a  column of $\\mathbf{A}$ and their cofactors equals $\\det(\\mathbf{A}),$ while the  sum of the products of the entries of a  column of $\\mathbf{A}$ and the cofactors of the entries of a different column equals zero$;$ that is$,$ \\begin{equation} \\label{eq:6.1.9} \\sum^n_{k=1} c_{ki}a_{kj}=\\left\\{\\casespace\\begin{array}{ll} \\det(\\mathbf{A}), &i=j,\\\\  0,&i\\ne j.\\end{array}\\right. \\end{equation} \\end{alist}"}
{"_id": "175", "title": "", "text": "Let $\\mathbf{A}$ be an $n\\times n$ matrix$.$ If $\\det(\\mathbf{A})=0,$ then $\\mathbf{A}$ is singular$.$ If $\\det(\\mathbf{A})\\ne0,$ then $\\mathbf{A}$ is nonsingular$,$ and $\\mathbf{A}$ has the unique inverse \\begin{equation} \\label{eq:6.1.10} \\mathbf{A}^{-1}=\\frac{1}{\\det(\\mathbf{A})}\\adj(\\mathbf{A}). \\end{equation}"}
{"_id": "176", "title": "", "text": "The system $\\eqref{eq:6.1.11}$ has a solution $\\mathbf{X}$ for any given $\\mathbf{Y}$ if and only if $\\mathbf{A}$ is nonsingular$.$ In this case$,$ the solution is unique and is given by $\\mathbf{X}=\\mathbf{A}^{-1}\\mathbf{Y}$."}
{"_id": "177", "title": "", "text": "If $\\mathbf{A}=[a_{ij}]$ is nonsingular$,$ then the solution of  the system \\begin{eqnarray*} a_{11}x_1+a_{12}x_2+\\cdots+a_{1n}x_n\\ar=y_1\\\\ a_{21}x_1+a_{22}x_2+\\cdots+a_{2n}x_n\\ar=y_2\\\\ &\\vdots& \\\\ a_{n1}x_1+a_{n2}x_2+\\cdots+a_{nn}x_n\\ar=y_n \\end{eqnarray*} $($or$,$ in matrix form$,$ $\\mathbf{AX}=\\mathbf{Y}$$)$ is given by $$ x_i=\\frac{D_i}{\\det(\\mathbf{A})},\\quad 1\\le i\\le n, $$ where $D_i$ is the determinant of the matrix obtained by replacing the $i$th column of $\\mathbf{A}$ with $\\mathbf{Y};$ thus$,$ $$ D_1=\\left|\\begin{array}{cccc} y_1&a_{12}&\\cdots&a_{1n}\\\\ y_2&a_{22}&\\dots&a_{2n}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ y_n&a_{n2}&\\cdots&a_{nn}\\end{array}\\right|,\\quad D_2=\\left|\\begin{array}{ccccc} a_{11}&y_1&a_{13}&\\cdots&a_{1n}\\\\ a_{21}&y_2&a_{23}&\\cdots&a_{2n}\\\\ \\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\ a_{n1}&y_n&a_{n3}&\\cdots&a_{nn}\\end{array}\\right|,\\quad\\cdots, $$ $$ D_n=\\left|\\begin{array}{cccc}  a_{11}&\\cdots&a_{1,n-1}&y_1\\\\ a_{21}&\\cdots&a_{2,n-1}&y_2\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ a_{n1}&\\cdots&a_{n,n-1}&y_n\\end{array}\\right|. $$"}
{"_id": "178", "title": "", "text": "The homogeneous system  $\\eqref{eq:6.1.12}$ of $n$ equations in $n$ unknowns has a nontrivial solution if and only if $\\det(\\mathbf{A})=0.$"}
{"_id": "179", "title": "", "text": "If $A_1,$ $A_2,$ \\dots$,$ $A_k$ are nonsingular $n\\times n$ matrices$,$ then so is $A_1A_2\\cdots A_k,$ and $$ (A_1A_2\\cdots A_k)^{-1}=A_k^{-1}A_{k-1}^{-1}\\cdots A_1^{-1}. $$"}
{"_id": "180", "title": "", "text": "Suppose that $\\mathbf{X}_0$ is in$,$ and a limit point of$,$ the domain of $\\mathbf{F}: \\R^n\\to\\R^m.$ Then $\\mathbf{F}$ is continuous at $\\mathbf{X}_0$ if and only if for each $\\epsilon>0$ there is a $\\delta>0$ such that \\begin{equation}\\label{eq:6.2.1} |\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)|<\\epsilon \\mbox{\\quad if \\quad} |\\mathbf{X}-\\mathbf{X}_0|<\\delta \\mbox{\\quad and \\quad} \\mathbf{X}\\in D_\\mathbf{F}. \\end{equation}"}
{"_id": "181", "title": "", "text": "A transformation $\\mathbf{F}=(f_1,f_2, \\dots,f_m)$ defined in a neighborhood of $\\mathbf{X}_0\\in\\R^n$  is differentiable at $\\mathbf{X}_0$ if and only if there is a constant $m\\times n$ matrix $\\mathbf{A}$ such that \\begin{equation}\\label{eq:6.2.2} \\lim_{\\mathbf{X}\\to\\mathbf{X}_0} \\frac{ \\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)-\\mathbf{A} (\\mathbf{X}-\\mathbf{X}_0)} {|\\mathbf{X}-\\mathbf{X}_0|}=\\mathbf{0}.  \\end{equation} If $\\eqref{eq:6.2.2}$ holds$,$ then $\\mathbf{A}$ is given uniquely by \\begin{equation}\\label{eq:6.2.3} \\mathbf{A}=\\left[\\frac{\\partial f_i(\\mathbf{X}_0)}{\\partial x_j}\\right]= \\left[\\begin{array}{cccc}\\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial x_1}}& \\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial x_2}}&\\cdots& \\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial x_n}}\\\\ [3\\jot] \\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_1}}& \\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_2}}& \\cdots&\\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_n}}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ \\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x_1}}& \\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x _2}}& \\cdots&\\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x_n}} \\end{array}\\right]. \\end{equation}"}
{"_id": "182", "title": "", "text": "If $\\mathbf{F}: \\R^n\\to\\R^m$ is differentiable at $\\mathbf{X}_0,$ then $\\mathbf{F}$ is continuous at~$\\mathbf{X}_0.$"}
{"_id": "183", "title": "", "text": "Let  $\\mathbf{F}=(f_1,f_2, \\dots,f_m):\\R^n\\to\\R^m,$ and suppose that the partial derivatives \\begin{equation}\\label{eq:6.2.7} \\frac{\\partial f_i}{\\partial x_j},\\quad 1\\le i\\le m,\\quad 1\\le j\\le n, \\end{equation} exist on a neighborhood of $\\mathbf{X}_0$ and are continuous at $\\mathbf{X}_0.$ Then $\\mathbf{F}$ is differentiable at $\\mathbf{X}_0.$"}
{"_id": "184", "title": "", "text": "Suppose that $\\mathbf{F}: \\R^n\\to\\R^m$ is differentiable at $\\mathbf{X}_0,$ $\\mathbf{G}:\\R^k\\to\\R^n$ is differentiable at $\\mathbf{U}_0,$ and $\\mathbf{X}_0=\\mathbf{G}(\\mathbf{U}_0).$ Then the composite function $\\mathbf{H}=\\mathbf{F}\\circ\\mathbf{G}:\\R^k\\to\\R^m,$ defined by $$ \\mathbf{H}(\\mathbf{U})=\\mathbf{F}(\\mathbf{G}(\\mathbf{U})), $$ is differentiable at $\\mathbf{U}_0.$ Moreover$,$ \\begin{equation}\\label{eq:6.2.22} \\mathbf{H}'(\\mathbf{U}_0)=\\mathbf{F}'(\\mathbf{G}(\\mathbf{U}_0)) \\mathbf{G}'(\\mathbf{U}_0) \\end{equation} and \\begin{equation}\\label{eq:6.2.23} d_{\\mathbf{U}_0}\\mathbf{H}=d_{\\mathbf{X}_0}\\mathbf{F}\\circ d_{\\mathbf{U}_0}\\mathbf{G}, \\end{equation} where $\\circ$ denotes composition$.$"}
{"_id": "185", "title": "", "text": "The linear transformation $$ \\mathbf{U}=\\mathbf{L}(\\mathbf{X})=\\mathbf{A}\\mathbf{X}\\quad  (\\R^n\\to \\R^n) $$ is invertible if and only if $\\mathbf{A}$ is nonsingular$,$ in which case $R(\\mathbf{L})= \\R^n$ and $$ \\mathbf{L}^{-1}(\\mathbf{U})=\\mathbf{A}^{-1}\\mathbf{U}. $$"}
{"_id": "186", "title": "", "text": "Suppose that $\\mathbf{F}: \\R^n\\to \\R^n$ is regular on an open set $S,$ and let $\\mathbf{G}=\\mathbf{F}^{-1}_S.$ Then $\\mathbf{F}(S)$ is open$,$ $\\mathbf{G}$ is continuously differentiable on $\\mathbf{F}(S),$ and $$ \\mathbf{G}'(\\mathbf{U})=(\\mathbf{F}'(\\mathbf{X}))^{-1}, \\mbox{\\quad where\\quad}\\mathbf{U}=\\mathbf{F}(\\mathbf{X}). $$ Moreover$,$ since $\\mathbf{G}$ is one-to-one on $\\mathbf{F}(S),$  $\\mathbf{G}$ is regular on $\\mathbf{F}(S).$"}
{"_id": "187", "title": "The Inverse Function Theorem", "text": "Let $\\mathbf{F}: \\R^n\\to \\R^n$ be continuously differentiable on an open set $S,$ and suppose that $J\\mathbf{F}(\\mathbf{X})\\ne0$ on $S.$ Then$,$ if $\\mathbf{X}_0\\in S,$ there is an open neighborhood $N$ of $\\mathbf{X}_0$ on which $\\mathbf{F}$ is regular$.$ Moreover$,$ $\\mathbf{F}(N)$ is open and $\\mathbf{G}= \\mathbf{F}^{-1}_N$ is continuously differentiable on $\\mathbf{F}(N),$ with $$ \\mathbf{G}'(\\mathbf{U})=\\left[\\mathbf{F}'(\\mathbf{X})\\right]^{-1}\\quad \\mbox{ $($where $\\mathbf{U}=\\mathbf{F}(\\mathbf{X})$$)$},\\quad \\mathbf{U}\\in\\mathbf{F}(N). $$"}
{"_id": "188", "title": "The Implicit Function Theorem", "text": "Suppose that  $\\mathbf{F}:\\R^{n+m}\\to \\R^m$ is continuously differentiable on an open set $S$ of $\\R^{n+m}$ containing $(\\mathbf{X}_0,\\mathbf{U}_0).$ Let $\\mathbf{F}(\\mathbf{X}_0,\\mathbf{U}_0)=\\mathbf{0},$ and suppose that  $\\mathbf{F}_\\mathbf{U}(\\mathbf{X}_0,\\mathbf{U}_0)$ is nonsingular$.$ Then there is a neighborhood $M$ of  $(\\mathbf{X}_0,\\mathbf{U}_{0}),$  contained in $S,$ on which  $\\mathbf{F}_\\mathbf{U}(\\mathbf{X},\\mathbf{U})$  is nonsingular  and a neighborhood $N$ of $\\mathbf{X}_0$ in $\\R^n$ on which a unique  continuously differentiable  transformation $\\mathbf{G}: \\R^n\\to \\R^m$ is defined$,$ such that $\\mathbf{G}(\\mathbf{X}_0)=\\mathbf{U}_0$ and \\begin{equation} \\label{eq:6.4.6} (\\mathbf{ X},\\mathbf{G}(\\mathbf{X}))\\in M\\mbox{\\quad and \\quad} \\mathbf{F}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))=0\\mbox{\\quad  if}\\quad\\mathbf{X}\\in N. \\end{equation} Moreover$,$ \\begin{equation} \\label{eq:6.4.7} \\mathbf{G}'(\\mathbf{X})=-[\\mathbf{F}_\\mathbf{U}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))]^{-1} \\mathbf{F}_\\mathbf{X}(\\mathbf{X},\\mathbf{G}(\\mathbf{X})),\\quad \\mathbf{X}\\in N. \\end{equation}"}
{"_id": "189", "title": "", "text": "If $f$ is unbounded on the nondegenerate rectangle $R$ in $\\R^n,$ then $f$ is not integrable on $R.$"}
{"_id": "190", "title": "", "text": "Let $f$ be bounded on a rectangle  $R$  and let $\\mathbf{P}$ be a partition of $R.$  Then \\begin{alist} \\item % (a)  The upper sum $S(\\mathbf{P})$ of $f$ over $\\mathbf{P}$ is the supremum of the set of all Riemann sums of $f$ over $\\mathbf{P}.$ \\item % (b)  The lower sum $s(\\mathbf{P})$ of $f$ over $\\mathbf{P}$ is the infimum  of the set of all Riemann sums of $f$ over $\\mathbf{P}.$ \\end{alist}"}
{"_id": "191", "title": "", "text": "If $f$ is bounded on a rectangle $R,$ then $$ \\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X} \\le\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}. $$"}
{"_id": "192", "title": "", "text": "If $f$ is integrable on a rectangle $R,$ then $$ \\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}= \\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X} =\\int_R f(\\mathbf{X})\\,d\\mathbf{X}. $$"}
{"_id": "193", "title": "", "text": "If $f$ is bounded on a rectangle $R$ and \\vspace{2pt} $$ \\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}= \\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=L, $$ \\vspace{2pt} then $f$ is integrable on $R,$ and \\vspace{2pt} $$ \\int_R f(\\mathbf{X})\\,d\\mathbf{X}=L. $$"}
{"_id": "194", "title": "", "text": "A bounded function $f$ is integrable on a rectangle $R$ if and only if $$ \\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}=\\overline{\\int_R}\\, f(\\mathbf{X})\\, d\\mathbf{X}. $$"}
{"_id": "195", "title": "", "text": "If $f$ is bounded on a rectangle $R,$ then $f$ is integrable on $R$ if and only if for every $\\epsilon>0$ there is a partition ${\\bf P}$ of $R$ such that $$ S({\\bf P})-s({\\bf P})<\\epsilon. $$"}
{"_id": "196", "title": "", "text": "If $f$ is continuous on a rectangle $R$ in $\\R^n,$ then $f$ is integrable on~$R.$"}
{"_id": "197", "title": "", "text": "Suppose that $f$ is bounded on a rectangle \\begin{equation}\\label{eq:7.1.30} R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n] \\end{equation} and continuous except on a subset $E$ of $R$ with zero content$.$ Then $f$ is integrable on $R.$"}
{"_id": "198", "title": "", "text": "Suppose that $f$ is bounded on a bounded set $S$ and continuous except on a subset $E$ of $S$ with zero content. Suppose also that $\\partial S$ has zero content$.$ Then $f$ is integrable on $S.$"}
{"_id": "199", "title": "", "text": "A differentiable surface in $\\R^n$ has zero content$.$"}
{"_id": "200", "title": "", "text": "Suppose that $S$ is a bounded set in $\\R^n,$ with boundary consisting of a finite number of differentiable surfaces$.$ Let $f$ be bounded on $S$ and continuous except on a set of zero content. Then $f$ is integrable on $S.$"}
{"_id": "201", "title": "", "text": "If $f$ and $g$ are integrable on $S,$ then so is $f+g,$ and $$ \\int_S(f+g)(\\mathbf{X})\\,d\\mathbf{X}=\\int_S f(\\mathbf{X})\\,d\\mathbf{X}+ \\int_S g(\\mathbf{X})\\,d\\mathbf{X}. $$"}
{"_id": "202", "title": "", "text": "If $f$ is integrable on $S$ and $c$ is a constant$,$ then $cf$ is integrable on $S,$ and $$ \\int_S(cf)(\\mathbf{X})\\,d\\mathbf{X}=c\\int_S f(\\mathbf{X})\\,d\\mathbf{X}. $$"}
{"_id": "203", "title": "", "text": "If $f$ and $g$ are integrable on $S$ and $f(\\mathbf{X})\\le g(\\mathbf{X})$ for $\\mathbf{X}$ in $S,$ then $$ \\int_S f(\\mathbf{X})\\,d\\mathbf{X}\\le\\int_S g(\\mathbf{X})\\,d\\mathbf{X}. $$"}
{"_id": "204", "title": "", "text": "If $f$ is integrable on $S,$ then so is $|f|,$ and $$ \\left|\\int_S f(\\mathbf{X})\\,d\\mathbf{X}\\right|\\le\\int_S |f(\\mathbf{X})|\\,d\\mathbf{X}. $$"}
{"_id": "205", "title": "", "text": "If $f$ and $g$ are integrable on $S,$ then so is the product $fg.$"}
{"_id": "206", "title": "", "text": "Suppose that $u$ is continuous and $v$ is integrable and nonnegative on a rectangle $R.$ Then $$ \\int_R u(\\mathbf{X})v(\\mathbf{X})\\,d\\mathbf{X}= u(\\mathbf{X}_0)\\int_R v(\\mathbf{X})\\,d\\mathbf{X} $$ for some $\\mathbf{X}_0$ in $R.$"}
{"_id": "207", "title": "", "text": "If $f$ is integrable on disjoint sets $S_1$ and $S_2,$ then $f$ is integrable on $S_1\\cup S_2,$ and \\begin{equation}\\label{eq:7.1.39} \\int_{S_1\\cup S_2} f(\\mathbf{X})\\,d\\mathbf{X}= \\int_{S_1} f(\\mathbf{X})\\,d\\mathbf{X}+ \\int_{S_2} f(\\mathbf{X})\\,d\\mathbf{X}. \\end{equation}"}
{"_id": "208", "title": "", "text": "$R= [a,b]\\times [c,d]$ and $$  F(y)=\\int_a^b f(x,y)\\,dx $$ exists for each $y$ in $[c,d].$ Then $F$ is integrable on $[c,d],$ and \\begin{equation}\\label{eq:7.2.1} \\int_c^d F(y)\\,dy=\\int_R f(x,y)\\,d(x,y); \\end{equation} that is$,$ \\begin{equation}\\label{eq:7.2.2} \\int_c^d dy\\int_a^b f(x,y)\\,dx=\\int_R f(x,y)\\,d(x,y). \\end{equation}"}
{"_id": "209", "title": "", "text": "Let $I_1,$ $I_2,$ \\dots$,$ $I_n$ be closed intervals and suppose that  $f$ is integrable on $R=I_1\\times I_2\\times\\cdots\\times I_n.$ Suppose that  there is an integer $p$ in $\\{1,2, \\dots,n-1\\}$ such that $$ F_p(x_{p+1},x_{p+2}, \\dots,x_n)=\\int_{I_1\\times I_2\\times\\cdots\\times I_p} f(x_1,x_2, \\dots,x_n)\\,d(x_1,x_2, \\dots,x_p) $$ exists for each $(x_{p+1},x_{p+2}, \\dots,x_n)$ in $I_{p+1}\\times I_{p+2}\\times\\cdots\\times I_n.$ Then $$ \\int_{I_{p+1}\\times I_{p+2}\\times\\cdots\\times I_n} F_p(x_{p+1}, x_{p+2}, \\dots,x_n)\\,d(x_{p+1},x_{p+2}, \\dots,x_n) $$ exists and equals $\\int_R f(\\mathbf{X})\\,d\\mathbf{X}$."}
{"_id": "210", "title": "", "text": "Let $I_j=[a_j,b_j],$ $1\\le j\\le n$, and suppose that $f$ is integrable on $R=I_1\\times I_2 \\times\\cdots\\times I_n.$ Suppose also that  the integrals $$ F_p(x_{p+1}, \\dots,x_n)=\\int_{I_1\\times I_2\\cdots\\times I_p} f(\\mathbf{X}) \\,d(x_1,x_2, \\dots,x_p),\\quad1\\le p\\le n-1, $$ exist for all $$ (x_{p+1}, \\dots,x_n)\\mbox{\\quad in\\quad} I_{p+1}\\times\\cdots\\times I_n. $$ Then the iterated integral $$ \\int^{b_n}_{a_n} dx_n\\int^{b_{n-1}}_{a_{n-1}} dx_{n-1}\\cdots \\int^{b_2}_{a_2} dx_2\\int^{b_1}_{a_1} f(\\mathbf{X})\\,dx_1 $$ exists and equals $\\int_R f(\\mathbf{X})\\,d\\mathbf{X}.$"}
{"_id": "211", "title": "", "text": "If $f$ is continuous on $$ R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n], $$ then $\\int_R f(\\mathbf{X})\\,d\\mathbf{X}$ can be evaluated by iterated integrals in any of the $n!$ ways indicated in $\\eqref{eq:7.2.16}.$"}
{"_id": "212", "title": "", "text": "If $f$ is integrable on the set $S$ in $\\eqref{eq:7.2.17}$ and the integral $\\eqref{eq:7.2.19}$ exists for $c\\le y\\le d,$ then \\begin{equation}\\label{eq:7.2.20} \\int_S f(x,y) \\,d(x,y)=\\int_c^d dy\\int^{v(y)}_{u(y)} f(x,y)\\,dx. \\end{equation}"}
{"_id": "213", "title": "", "text": "Suppose that  $f$ is integrable on $$ S=\\set{(x,y,z)}{u_1(y,z)\\le x\\le v_1(y,z),\\ u_2(z)\\le y\\le v_2(z),\\ c\\le z\\le d}, $$ and let $$ S(z)=\\set{(x,y)}{u_1(y,z)\\le x\\le v_1(y,z),\\ u_2(z)\\le y\\le v_2(z)} $$ for each $z$ in $[c,d].$  Then $$ \\int_S f(x,y,z)\\,d(x,y,z)=\\int_c^d dz\\int^{v_2(z)}_{u_2(z)} dy \\int^{v_1(y,z)}_{u_1(y,z)} f(x,y,z)\\,dx, $$ provided that $$ \\int^{v_1(y,z)}_{u_1(y,z)} f(x,y,z)\\,dx $$ exists for all $(y,z)$ such that $$ c\\le z\\le d\\mbox{\\quad and\\quad} u_2(z)\\le y\\le v_2(z), $$ and $$ \\int_{S(z)} f(x,y,z)\\,d(x,y) $$ exists for all $z$ in $[c,d].$"}
{"_id": "214", "title": "", "text": "A bounded set $S$ is Jordan measurable if and only if the boundary of $S$ has zero content$.$"}
{"_id": "215", "title": "", "text": "Suppose that  $\\mathbf{G}:\\R^n\\to \\R^n$ is regular on a compact Jordan measurable set $S.$ Then $\\mathbf{G}(S)$ is compact and Jordan measurable$.$"}
{"_id": "216", "title": "", "text": "If $S$ is a compact Jordan measurable subset  of $\\R^n$ and $\\mathbf{L}:\\R^n\\to \\R^n$ is the invertible linear transformation $\\mathbf{X}=\\mathbf{L}(\\mathbf{Y})=\\mathbf{AY},$ then \\begin{equation}\\label{eq:7.3.14} V(\\mathbf{L}(S))=|\\det(\\mathbf{A})| V(S). \\end{equation}"}
{"_id": "217", "title": "", "text": "\\E^n\\to \\R^n$ is regular on a compact Jordan measurable set $S$ and $f$ is continuous on $\\mathbf{G}(S).$ Then \\begin{equation}\\label{eq:7.3.28} \\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}= \\int_S f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}. \\end{equation}"}
{"_id": "218", "title": "", "text": "Suppose that  $\\mathbf{G}: \\E^n\\to \\R^n$ is continuously differentiable on a bounded open set $N$ containing the compact Jordan measurable set $S,$ and regular on $S^0.$ Suppose   also that $\\mathbf{G}(S)$ is Jordan measurable$,$ $f$ is continuous on $\\mathbf{G}(S),$ and $G(C)$ is Jordan measurable for every cube $C\\subset N$. Then \\begin{equation}\\label{eq:7.3.50} \\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}= \\int_S f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}. \\end{equation}"}
{"_id": "219", "title": "", "text": "If $(A,N)$ is a normed vector space$,$ then \\begin{equation} \\label{eq:8.1.1} \\rho(x,y)=N(x-y) \\end{equation} is a metric on $A.$"}
{"_id": "220", "title": "", "text": "If $x$ and $y$ are vectors in  a normed vector space $(A,N),$ then \\begin{equation} \\label{eq:8.1.2} |N(x)-N(y)|\\le N(x-y). \\end{equation}"}
{"_id": "221", "title": "", "text": "If $\\mathbf{X}\\in\\R^n$ and $p_2>p_1\\ge1,$ then \\begin{equation} \\label{eq:8.1.12} \\|\\mathbf{X}\\|_{p_2}\\le\\|\\mathbf{X}\\|_{p_1}; \\end{equation} moreover, \\begin{equation} \\label{eq:8.1.13} \\lim_{p\\to\\infty}\\|\\mathbf{X}\\|_{p}=\\max\\set{|x_i|}{1\\le i\\le n}. \\end{equation}"}
{"_id": "222", "title": "", "text": "\\begin{alist} \\item % (a)   The union of open sets is open. \\item % (b)  The intersection of closed sets is closed. \\end{alist}"}
{"_id": "224", "title": "", "text": "\\begin{alist} \\item  % (a) The limit of a convergent sequence is unique$.$ \\item % (b) If $\\lim_{n\\to\\infty}u_n=u,$  then every subsequence of $\\{u_n\\}$  converges to $u.$ \\end{alist}"}
{"_id": "225", "title": "", "text": "If a sequence $\\{u_n\\}$ in a metric space $(A,\\rho)$ is convergent$,$ then it is a Cauchy sequence."}
{"_id": "226", "title": "The Principle of Nested Sets", "text": "A metric space $(A,\\rho)$ is complete if and only if every nested sequence $\\{T_n\\}$ of nonempty closed subsets of $A$ such that  $\\lim_{n\\to\\infty}d(T_n)=0$ has a nonempty intersection$.$"}
{"_id": "227", "title": "", "text": "If $\\rho$ and $\\sigma$ are equivalent  metrics on a set $A,$ then  $(A,\\rho)$ and $(A,\\sigma)$ have the same open sets."}
{"_id": "228", "title": "", "text": "Any two norms  $N_1$ and $N_2$ on $\\R^n$ induce equivalent metrics on~$\\R^n.$"}
{"_id": "229", "title": "", "text": "Suppose that $\\rho$ and $\\sigma$ are equivalent metrics on $A.$ Then \\begin{alist} \\item % (a) A sequence $\\{u_n\\}$ converges to $u$ in $(A,\\rho)$ if and only if it converges to $u$ in~$(A,\\sigma).$ \\item % (a) A sequence $\\{u_n\\}$ is a Cauchy sequence in  $(A,\\rho)$ if and only if it is a Cauchy sequence in $(A,\\sigma).$ \\item % (b) $(A,\\rho)$ is complete if and only if $(A,\\sigma)$ is complete$.$ \\end{alist}"}
{"_id": "230", "title": "", "text": "An infinite subset $T$ of $A$ is compact if and only if every infinite subset of $T$ has a limit point in $T.$"}
{"_id": "231", "title": "", "text": "A subset $T$ of a metric $A$ is compact if and only if every infinite sequence $\\{t_n\\}$ of members of  $T$  has a subsequence that converges to a member of $T.$"}
{"_id": "232", "title": "", "text": "If  $T$ is  compact$,$  then every Cauchy sequence $\\{t_n\\}_{n=1}^\\infty$ in $T$ converges to a limit in $T.$"}
{"_id": "233", "title": "", "text": "If $T$ is compact$,$ then $T$ is closed and bounded."}
{"_id": "234", "title": "", "text": "If $T$  is compact$,$ then $T$ is totally bounded."}
{"_id": "235", "title": "", "text": "If $(A,\\rho)$ is complete and $T$  is closed and totally bounded$,$ then $T$ is compact."}
{"_id": "236", "title": "", "text": "A nonempty subset $T$ of $C[a,b]$ is compact if and only if it is closed$,$ uniformly bounded$,$ and equicontinuous."}
{"_id": "237", "title": "", "text": "Suppose that  ${\\mathcal F}$ is an infinite uniformly bounded and equicontinuous family of functions on $[a,b].$  Then there is a sequence $\\{f_n\\}$ in ${\\mathcal F}$ that converges uniformly to a continuous function  on $[a,b].$"}
{"_id": "238", "title": "", "text": "Suppose that $\\widehat u\\in\\overline D_f.$ Then \\begin{equation} \\label{eq:8.3.3} \\lim_{u\\to \\widehat u}f(u)=\\widehat v \\end{equation} if and only if \\begin{equation} \\label{eq:8.3.4} \\lim_{n\\to\\infty}f(u_n)=\\widehat v \\end{equation} for every sequence $\\{u_n\\}$ in $D_f$ such that \\begin{equation} \\label{eq:8.3.5} \\lim_{n\\to\\infty}u_n=\\widehat u. \\end{equation}"}
{"_id": "239", "title": "", "text": "A function $f$ is continuous at $\\widehat u$ if and only if $$ \\lim_{u\\to\\widehat u} f(u)=f(\\widehat u). $$"}
{"_id": "240", "title": "", "text": "A function $f$ is continuous at $\\widehat u$ if and only if $$ \\lim_{n\\to\\infty} f(u_n)=f(\\widehat u) $$ whenever $\\{u_n\\}$ is a sequence in $D_f$  that converges to $\\widehat u$."}
{"_id": "241", "title": "", "text": "If $f$ is continuous on a compact set $T,$ then $f(T)$ is compact."}
{"_id": "242", "title": "", "text": "If $f$ is continuous on a compact set $T,$ then $f$ is uniformly continuous on $T$."}
{"_id": "243", "title": "Contraction Mapping Theorem", "text": "If $f$ is a contraction of a complete metric space $(A,\\rho),$ then the equation \\begin{equation} \\label{eq:8.3.8} f(u)=u \\end{equation} has a unique solution$.$"}
{"_id": "244", "title": "", "text": "If $f$ is differentiable at $x_0,$ then \\begin{equation}\\label{eq:2.3.3} f(x)=f(x_0)+[f'(x_0)+E(x)](x-x_0), \\end{equation} where $E$ is defined on a neighborhood of $x_0$ and $$ \\lim_{x\\to x_0} E(x)=E(x_0)=0. $$"}
{"_id": "245", "title": "", "text": "If $f^{(n)}(x_0)$ exists$,$ then \\begin{equation}\\label{eq:2.5.7} f(x)=\\sum_{r=0}^n\\frac{f^{(r)}(x_0)}{ r!} (x-x_0)^r+E_n(x)(x-x_0)^n, \\end{equation} where $$ \\lim_{x\\to x_0} E_n(x)=E_n(x_0)=0. $$"}
{"_id": "246", "title": "", "text": "Suppose that \\begin{equation} \\label{eq:3.2.1} |f(x)|\\le M,\\quad a\\le x\\le b, \\end{equation} and let $P'$ be a partition of $[a,b]$ obtained by adding $r$ points to a partition $P=\\{x_0,x_1, \\dots,x_n\\}$ of $[a,b].$  Then \\begin{eqnarray} S(P)\\ge S(P')\\ar\\ge S(P)-2Mr\\|P\\|\\label{eq:3.2.2}\\\\ \\arraytext{and}\\nonumber\\\\ s(P)\\le s(P')\\ar\\le s(P)+2Mr\\|P\\|\\label{eq:3.2.3}. \\end{eqnarray}"}
{"_id": "247", "title": "", "text": "If $f$ is bounded on $[a,b]$ and  $\\epsilon>0,$   there is a $\\delta>0$ such that \\begin{equation} \\label{eq:3.2.12} \\overline{\\int_a^b}f(x)\\,dx\\le S(P)<\\overline{\\int_a^b}f(x)\\,dx+\\epsilon \\end{equation} and $$ \\underline{\\int_a^b} f(x)\\,dx\\ge s(P)>\\underline{\\int_a^b} f(x)\\,dx-\\epsilon $$ if $\\|P\\|<\\delta$."}
{"_id": "248", "title": "", "text": "If $w_f(x)<\\epsilon$ for $a\\le x \\le b,$ then there is a $\\delta>0$ such that $W_f[a_1,b_1]\\le\\epsilon,$ provided that $[a_1,b_1]\\subset [a,b]$ and $b_1-a_1<\\delta.$"}
{"_id": "249", "title": "", "text": "Let $f$ be bounded on $[a,b]$ and define $$ E_\\rho=\\set{x\\in [a,b]}{w_f(x)\\ge\\rho}. $$ Then $E_\\rho$ is closed$,$ and  $f$ is integrable on $[a,b]$ if and only if for every pair of positive numbers $\\rho$ and $\\delta,$ $E_\\rho$ can be covered by finitely many open intervals $I_1,$ $I_2, $\\dots$,$ $I_p$ such that \\begin{equation} \\label{eq:3.5.3} \\sum_{j=1}^p L(I_j)<\\delta. \\end{equation}"}
{"_id": "250", "title": "", "text": "Suppose that for $n$ sufficiently large  $($that is$,$ for $n \\ge\\mbox{some integer }N$$)$  the terms of $\\sum_{n=k}^\\infty a_n$ satisfy  some condition that implies convergence of an infinite series$.$ Then $\\sum_{n=k}^\\infty a_n$ converges$.$ Similarly, suppose that for $n$  sufficiently large  the terms $\\sum_{n=k}^\\infty a_n$ satisfy  some condition that implies divergence of an infinite series$.$ Then $\\sum_{n=k}^\\infty a_n$ diverges$.$"}
{"_id": "251", "title": "", "text": "If $g$ and $h$ are defined on $S,$ then \\begin{eqnarray*} \\|g+h\\|_S\\ar\\le\\|g\\|_S+\\|h\\|_S\\\\ \\arraytext{and}\\\\ \\|gh\\|_S\\ar\\le\\|g\\|_S\\|h\\|_S. \\end{eqnarray*} Moroever$,$ if either $g$ or $h$ is bounded on $S,$ then $$ \\|g-h\\|_S\\ge\\left|\\|g\\|_S-\\|h\\|_S\\|\\right|. $$"}
{"_id": "252", "title": "", "text": "If $\\mathbf{X}$ and $\\mathbf{Y}$ are any two vectors in $\\R^n,$ then \\begin{equation} \\label{eq:5.1.3} |\\mathbf{X}\\cdot\\mathbf{Y}|\\le |\\mathbf{X}|\\,|\\mathbf{Y}|, \\end{equation} with equality  if and only if one of the vectors is a scalar multiple of the other$.$"}
{"_id": "253", "title": "", "text": "If $\\mathbf{X}_1$ and $\\mathbf{X}_2$ are in $S_r(\\mathbf{X}_0)$ for some $r>0$, then so is every point on the line segment from $\\mathbf{X}_1$ to $\\mathbf{X}_2.$"}
{"_id": "254", "title": "", "text": "If $f$ is differentiable at $\\mathbf{X}_0,$ then $$ f(\\mathbf{X})-f(\\mathbf{X}_0)=(d_{\\mathbf{X}_0}f)(\\mathbf{X}-\\mathbf{X}_0) +E(\\mathbf{X})|\\mathbf{X}-\\mathbf{X}_0|, $$ where $E$  is defined in a neighborhood of $\\mathbf{X}_0$ and $$ \\lim_{\\mathbf{X}\\to\\mathbf{X}_0}E(\\mathbf{X})=E(\\mathbf{X}_0)=0. $$"}
{"_id": "255", "title": "", "text": "Suppose that  $\\mathbf{G}=(g_1,g_2, \\dots,g_n)$ is differentiable at $$  \\mathbf{U}_0=(u_{10}, u_{20}, \\dots,u_{m0}), $$  and  define $$ M=\\left(\\sum_{i=1}^n\\sum_{j=1}^m\\left(\\frac{\\partial g_i(\\mathbf{U}_0} {\\partial u_j}\\right)^2\\right)^{1/2}. $$ Then$,$ if $\\epsilon>0,$ there is a $\\delta>0$ such that $$ \\frac{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|} {|\\mathbf{U}-\\mathbf{U}_{0}|} <M+\\epsilon\\mbox{\\quad if \\quad}0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta. $$"}
{"_id": "256", "title": "", "text": "Suppose that $\\mathbf{F}:\\R^n\\to\\R^m$ is continuously differentiable on a neighborhood $N$ of $\\mathbf{X}_0.$ Then$,$ for every $\\epsilon>0,$ there is a $\\delta>0$ such that \\begin{equation}\\label{eq:6.2.8} |\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})|< (\\|\\mathbf{F}'(\\mathbf{X}_{0})\\| +\\epsilon) |\\mathbf{X}-\\mathbf{Y}| \\mbox{\\quad if\\quad}\\mathbf{A},\\mathbf{Y}\\in B_\\delta (\\mathbf{X}_0). \\end{equation}"}
{"_id": "257", "title": "", "text": "Suppose that $\\mathbf{F}:\\R^n\\to\\R^n$ is continuously differentiable on a neighborhood of $\\mathbf{X}_0$  and $\\mathbf{F}'(\\mathbf{X}_0)$ is nonsingular$.$ Let \\begin{equation}\\label{eq:6.2.14} r=\\frac{1}{\\|(\\mathbf{F}'(\\mathbf{X}_0))^{-1}\\|}. \\end{equation} Then$,$ for every $\\epsilon>0,$ there is a $\\delta>0$ such that \\begin{equation}\\label{eq:6.2.15} |\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})|\\ge (r-\\epsilon) |\\mathbf{X}-\\mathbf{Y}|\\mbox{\\quad if\\quad} \\mathbf{X},\\mathbf{Y}\\in B_\\delta(\\mathbf{X}_{0}). \\end{equation}"}
{"_id": "258", "title": "", "text": "If $\\mathbf{F}:\\R^n\\to\\R^m$ is continuously differentiable on an open set containing a compact set $D,$ then there is a constant $M$ such that \\begin{equation}\\label{eq:6.2.18} |\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X})|\\le M|\\mathbf{Y}-\\mathbf{X}| \\mbox{\\quad if\\quad}\\mathbf{X},\\mathbf{Y}\\in D. \\end{equation}"}
{"_id": "259", "title": "", "text": "Suppose that $|f(\\mathbf{X})|\\le M$ if $\\mathbf{X}$ is in the rectangle $$ R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n]. $$ Let ${\\bf P}=P_1\\times P_2\\times\\cdots\\times P_n$ and ${\\bf P}'= P_1'\\times P_2'\\times\\cdots\\times P_n'$ be partitions of $R,$ where $P_j'$ is obtained by adding $r_j$ partition points to $P_j,$ $1\\le j\\le n.$ Then \\begin{equation}\\label{eq:7.1.16} S({\\bf P})\\ge S({\\bf P}')\\ge S({\\bf P})-2MV(R)\\left(\\sum_{j=1}^n \\frac{r_j}{ b_j-a_j}\\right)\\|{\\bf P}\\| \\end{equation} and \\begin{equation}\\label{eq:7.1.17} s({\\bf P})\\le s({\\bf P}')\\le s({\\bf P})+2MV(R)\\left(\\sum_{j=1}^n \\frac{r_j }{ b_j-a_j}\\right)\\|{\\bf P}\\|. \\end{equation}"}
{"_id": "260", "title": "", "text": "If $f$ is bounded on a rectangle $R$ and $\\epsilon>0,$  there is  a $\\delta>0$ such that \\vspace{4pt} $$ \\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\le S({\\bf P})<\\overline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}+\\epsilon $$ \\vspace{4pt} and \\vspace{4pt} $$ \\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}\\ge s({\\bf P})> \\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}-\\epsilon $$ \\vspace{4pt} if $\\|{\\bf P}\\|<\\delta.$"}
{"_id": "261", "title": "", "text": "The union of finitely many sets with zero content has zero content$.$"}
{"_id": "262", "title": "", "text": "Suppose that $S$ is contained in a bounded set $T$ and $f$ is integrable on $S.$ Then  $f_S$ $($see $\\eqref{eq:7.1.36})$ is integrable on $T,$ and $$ \\int_T f_S(\\mathbf{X})\\,d\\mathbf{X}=\\int_S f(\\mathbf{X})\\,d\\mathbf{X}. $$"}
{"_id": "263", "title": "", "text": "Suppose that  $K$ is a bounded set with zero content and $\\epsilon,$ $\\rho>0.$  Then there are cubes $C_1,$ $C_2,$ \\dots$,$ $C_r$ with edge lengths $<\\rho$ such that  $C_j\\cap K\\ne\\emptyset,$ $1\\le j\\le r,$ \\begin{equation}\\label{eq:7.3.5} K\\subset\\bigcup_{j=1}^r C_j, \\end{equation} and $$ \\sum_{j=1}^r V(C_j)<\\epsilon. $$"}
{"_id": "264", "title": "", "text": "Suppose that  $\\mathbf{G}: \\R^n\\to \\R^n$ is continuously differentiable on a bounded  open set $S,$ and let $K$ be a closed subset of $S$ with zero content$.$ Then $\\mathbf{G}(K)$ has zero content."}
{"_id": "265", "title": "", "text": "A nonsingular $n\\times n$ matrix $\\mathbf{A}$ can be written as \\begin{equation}\\label{eq:7.3.10} \\mathbf{A}=\\mathbf{E}_k\\mathbf{E}_{k-1}\\cdots\\mathbf{E}_1, \\end{equation} where each $\\mathbf{E}_i$ is a matrix that can be obtained from the $n\\times n$ identity matrix $\\mathbf{I}$ by one of the following operations$:$ \\begin{alist} \\item % (a) interchanging two rows of $\\mathbf{I};$ \\item % (b) multiplying a row of $\\mathbf{I}$ by a nonzero constant$;$ \\item % (c) adding a multiple of one row of $\\mathbf{I}$ to another$.$ \\end{alist}"}
{"_id": "266", "title": "", "text": "Suppose that  $\\mathbf{G}:\\E^n\\to \\R^n$ is regular  on a cube $C$ in $\\E^n,$ and let $\\mathbf{A}$ be a nonsingular $n\\times n$ matrix$.$ Then \\begin{equation}\\label{eq:7.3.29} V(\\mathbf{G}(C))\\le |\\det(\\mathbf{A})|\\left[\\max \\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C} \\right]^n V(C). \\end{equation}"}
{"_id": "267", "title": "", "text": "If $\\mathbf{G}:\\E^n\\rightarrow \\R^n$  is regular on a cube $C$ in $\\R^n,$ then \\begin{equation}\\label{eq:7.3.32} V(\\mathbf{G}(C))\\le\\int_C |JG(\\mathbf{Y})|\\,d\\mathbf{Y}. \\end{equation}"}
{"_id": "268", "title": "", "text": "Suppose that  $S$ is Jordan measurable and $\\epsilon,$  $\\rho>0.$  Then there are cubes $C_1,$ $C_2,$ \\dots$,$ $C_r$ in $S$ with edge lengths $<\\rho,$ such that  $C_j\\subset  S,$ $1\\le j\\le r,$ $C_i^0\\cap C_j^0=\\emptyset$ if $i\\ne j,$ and \\begin{equation} \\label{eq:7.3.35} V(S)\\le\\sum_{j=1}^r V(C_j)+\\epsilon. \\end{equation}"}
{"_id": "269", "title": "", "text": "Suppose that  $\\mathbf{G}: \\E^n\\to \\R^n$ is regular on a compact Jordan measurable set $S$ and $f$ is continuous and nonnegative on $\\mathbf{G}(S).$ Let \\begin{equation}\\label{eq:7.3.37} Q(S)=\\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}-\\int_S  f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}. \\end{equation} Then $Q(S)\\le0.$"}
{"_id": "270", "title": "", "text": "Under the assumptions of Lemma~$\\ref{thmtype:7.3.13},$ $Q(S)\\ge0.$"}
{"_id": "271", "title": "", "text": "Suppose that  $\\mu_1,$ $\\mu_2,$ \\dots$,$ $\\mu_n$ and $\\nu_1,$ $\\nu_2,$ \\dots$,$ $\\nu_n$ are nonnegative  numbers$.$ Let $p>1$ and $q=p/(p-1);$ thus$,$ \\begin{equation} \\label{eq:8.1.5} \\frac{1}{p}+\\frac{1}{q}=1. \\end{equation}  Then \\begin{equation} \\label{eq:8.1.6} \\sum_{i=1}^n \\mu_i\\nu_i\\le\\left(\\sum_{i=1}^n\\mu_i^p\\right)^{1/p} \\left(\\sum_{i=1}^n \\nu_i^q\\right)^{1/q}. \\end{equation}"}
{"_id": "272", "title": "", "text": "Suppose that  $u_1,$ $u_2,$ \\dots$,$ $u_n$ and $v_1,$ $v_2,$ \\dots$,$ $v_n$ are nonnegative  numbers and $p>1.$ Then \\begin{equation} \\label{eq:8.1.8} \\left(\\sum_{i=1}^n(u_i+v_i)^p\\right)^{1/p} \\le\\left(\\sum_{i=1}^n u_i^p\\right)^{1/p} +\\left(\\sum_{i=1}^n v_i^p\\right)^{1/p}. \\end{equation}"}
{"_id": "273", "title": "", "text": "If $a$ and $b$ are any two real numbers$,$ then \\begin{equation} \\label{eq:1.1.4} |a-b|\\ge\\big||a|-|b|\\big| \\end{equation} and \\begin{equation} \\label{eq:1.1.5} |a+b|\\ge\\big||a|-|b|\\big|. \\end{equation}"}
{"_id": "275", "title": "", "text": "If $f$ is continuous on a set $T,$ then  $f$ is  uniformly continuous on any finite closed interval contained in $T.$"}
{"_id": "276", "title": "", "text": "If $f'$ is integrable on $[a,b],$ then $$ \\int_a^b f'(x)\\,dx=f(b)-f(a). $$"}
{"_id": "277", "title": "", "text": "If $\\sum a_n$ converges$,$ then $\\lim_{n\\to\\infty}a_n=0.$"}
{"_id": "278", "title": "", "text": "If $\\sum a_n$ converges$,$ then for each $\\epsilon>0$ there is an integer $K$ such that $$ \\left|\\sum_{n=k}^\\infty a_n\\right|<\\epsilon\\mbox{\\quad if\\quad} k\\ge K; $$ that is$,$ $$ \\lim_{k\\to\\infty}\\sum_{n=k}^\\infty a_n=0. $$"}
{"_id": "279", "title": "", "text": "Suppose that  $a_n\\ge0$ and $b_n>0$ for $n\\ge k,$ and $$ \\lim_{n\\to\\infty}\\frac{a_n}{ b_n}=L, $$ where $0<L<\\infty.$ Then $\\sum a_n$ and $\\sum b_n$ converge or diverge together$.$"}
{"_id": "280", "title": "", "text": "Suppose that  $a_n>0\\ (n\\ge k)$ and $$ \\lim_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}=L. $$ \\vskip-1em Then \\begin{alist} \\item % (a)  $\\sum a_n<\\infty$ if  $L<1.$ \\item % (b)  $\\sum a_n=\\infty$  if  $L>1.$ \\end{alist} The test is inconclusive if $L=1.$"}
{"_id": "281", "title": "", "text": "The series $\\sum a_nb_n$ converges if $a_{n+1}\\le a_n$ for $n\\ge k,$ $\\lim_{n\\to\\infty}a_n=0,$ and $$ |b_k+b_{k+1}+\\cdots+b_n|\\le M,\\quad n\\ge k, $$ for some constant $M.$"}
{"_id": "282", "title": "Alternating Series Test", "text": "The series $\\sum (-1)^na_n$ converges if $0\\le a_{n+1}\\le a_n$ and $\\lim_{n\\to\\infty} a_n=0.$"}
{"_id": "283", "title": "", "text": "If $\\{F_n\\}$ converges uniformly to $F$ on $S$ and each $F_n$ is continuous on $S,$ then so is $F;$ that is$,$ a uniform limit of continuous functions is continuous."}
{"_id": "284", "title": "", "text": "If $\\sum f_n$ converges uniformly on $S,$ then $\\lim_{n\\to\\infty}\\|f_n\\|_S=0.$"}
{"_id": "285", "title": "", "text": "The series $\\sum_{n=k}^\\infty  f_ng_n$ converges uniformly on $S$ if $$ f_{n+1}(x)\\le f_n(x),\\quad x\\in S,\\quad n\\ge k, $$ $\\{f_n\\}$ converges uniformly to zero on $S,$ and $$ \\|g_k+g_{k+1}+\\cdots+g_n\\|_S\\le M,\\quad n\\ge k, $$ for some constant $M.$"}
{"_id": "286", "title": "", "text": "If $$ f(x)=\\sum^\\infty_{n=0} a_n(x-x_0)^n,\\quad |x-x_0|<R, $$ then $$ a_n=\\frac{f^{(n)}(x_0)}{ n!}. $$"}
{"_id": "287", "title": "Uniqueness of Power Series", "text": "If \\begin{equation}\\label{eq:4.5.13} \\sum^\\infty_{n=0} a_n(x-x_0)^n=\\sum^\\infty_{n=0} b_n(x-x_0)^n \\end{equation} for all $x$ in some interval $(x_0-r,x_0+r),$ then \\begin{equation}\\label{eq:4.5.14} a_n=b_n,\\quad n\\ge0. \\end{equation}"}
{"_id": "288", "title": "", "text": "If $\\mathbf{X},$ $\\mathbf{Y},$ and $\\mathbf{Z}$ are  in $\\R^n,$ then $$ |\\mathbf{X}-\\mathbf{Z}|\\le |\\mathbf{X}-\\mathbf{Y}|+|\\mathbf{Y}-\\mathbf{Z}|. $$"}
{"_id": "289", "title": "", "text": "If $\\mathbf{X}$ and $\\mathbf{Y}$ are  in $\\R^n,$ then $$ |\\mathbf{X}-\\mathbf{Y}|\\ge\\left| |\\mathbf{X}|-|\\mathbf{Y}|\\right|. $$"}
{"_id": "290", "title": "", "text": "Under the assumptions of Theorem~$\\ref{thmtype:5.4.3},$ \\begin{equation} \\label{eq:5.4.8} \\frac{\\partial h(\\mathbf{U}_0)}{\\partial u_i}=\\sum_{j=1}^n \\frac{\\partial f(\\mathbf{X}_0) }{\\partial x_j} \\frac{\\partial g_j(\\mathbf{U}_0)}{\\partial u_i},\\quad 1\\le i \\le m. \\end{equation}"}
{"_id": "291", "title": "", "text": "If $f_{x_1},$ $f_{x_2},$ \\dots$,$ $f_{x_n}$ are identically zero in an open region $S$ of $\\R^n,$ then $f$ is constant in $S.$"}
{"_id": "292", "title": "", "text": "Suppose that  $f,$ $f_x,$ and $f_y$ are differentiable in a neigborhood of a critical point $\\mathbf{X}_0=(x_0,y_0)$ of $f$ and $f_{xx},$ $f_{yy},$ and $f_{xy}$ are continuous at $(x_0,y_0).$ Let $$ D=f_{xx}(x_0,y_0)f_{xy}(x_0,y_0)-f^2_{xy}(x_0,y_0). $$ Then \\begin{alist} \\item % (a)  $(x_0,y_0)$ is a local extreme point of $f$ if $D>0;$ $(x_0,y_0)$ is a local minimum point if $f_{xx}(x_0,y_0)>0$, or a local maximum point if $f_{xx}(x_0,y_0)<0.$ \\item % (b) $(x_0,y_0)$ is not a local extreme point of $f$ if $D<0.$ \\end{alist}"}
{"_id": "293", "title": "", "text": "If $\\mathbf{F}$ is continuously differentiable on a neighborhood  of $\\mathbf{X}_0$ and $J\\mathbf{F}(\\mathbf{X}_0)\\ne 0,$ then there is an open neighborhood $N$ of $\\mathbf{X}_0$ on which the conclusions of Theorem~$\\ref{thmtype:6.3.4}$ hold$.$"}
{"_id": "294", "title": "", "text": "Suppose that  $f:\\R^{n+1}\\to \\R$ is continuously differentiable on an open set containing $(\\mathbf{X}_0,u_0),$ with $f(\\mathbf{X}_0,u_0)=0$ and $f_u(\\mathbf{X}_0,u_0)\\ne0$. Then there is a neighborhood $M$ of $(\\mathbf{X}_0,u_0),$ contained in $S,$ and a neighborhood $N$ of $\\mathbf{X}_0$ in $\\R^n$ on which is defined a unique continuously differentiable function $u=u(\\mathbf{X}):\\R^n\\to \\R$ such that $$ (\\mathbf{X},u(\\mathbf{X}))\\in M\\mbox{\\quad and \\quad}  f_u(\\mathbf{X},u(\\mathbf{X}))\\ne0,\\quad\\mathbf{X}\\in N, $$ $$ u(\\mathbf{X}_0)=u_0, \\mbox{\\quad and \\quad} f(\\mathbf{X},u(\\mathbf{X}))=0,\\quad\\mathbf{X}\\in N. $$ The partial derivatives of $u$ are given by $$ u_{x_i}(\\mathbf{X})=-\\frac{f_{x_i}(\\mathbf{X},u(\\mathbf{X}))}{ f_u(\\mathbf{X},u(\\mathbf{X}))},\\quad 1\\le i\\le n. $$"}
{"_id": "295", "title": "", "text": "Suppose that  $f$ is integrable on sets $S_1$ and $S_2$ such that $S_1\\cap S_2$ has zero content$.$ Then $f$ is integrable on $S_1\\cup S_2,$ and $$ \\int_{S_1\\cup S_2} f(\\mathbf{X})\\,d\\mathbf{X}= \\int_{S_1} f(\\mathbf{X})\\,d\\mathbf{X}+ \\int_{S_2} f(\\mathbf{X})\\,d\\mathbf{X}. $$"}
{"_id": "296", "title": "", "text": "If $f$ is integrable on $[a,b] \\times [c,d],$ then $$ \\int_a^b dx\\int_c^d f(x,y)\\,dy=\\int_c^d dy\\int_a^b f(x,y)\\,dx, $$ provided that $\\int_c^d f(x,y)\\,dy$ exists for $a\\le x\\le b$ and $\\int_a^b f(x,y)\\,dx$ exists for $c\\le y\\le d.$ In particular$,$ these hypotheses hold if $f$ is continuous on $[a,b]\\times [c,d].$"}
{"_id": "297", "title": "", "text": "If $f$ is bounded and continuous on a bounded Jordan measurable set $S,$ then $f$ is integrable on $S.$"}
{"_id": "298", "title": "", "text": "A set $D$  is  {\\it dense in the reals\\/} if every open interval $(a,b)$  contains a member of $D$."}
{"_id": "299", "title": "", "text": "Let $S$ and $T$ be sets. \\begin{alist} \\item % (a) $S$ {\\it contains\\/} $T$, and we write $S\\supset T$ or $T\\subset S$, if every member of $T$ is also in $S$. In this case, $T$ is a {\\it subset\\/} of $S$. \\item % (b)  $S-T$ is the set of elements that are in $S$ but not in $T$. \\item % (c) $S$ {\\it equals\\/} $T$, and we write $S=T$, if $S$ contains $T$ and $T$ contains $S$; thus, $S=T$ if and only if $S$ and $T$ have the same members. \\newpage \\item % (d)  $S$ {\\it strictly contains\\/} $T$ if $S$ contains $T$ but $T$ does not contain $S$; that is, if every member of  $T$ is also in $S$, but at least one member of $S$ is not in $T$ (Figure~\\ref{figure:1.3.1}). \\item % (e) The {\\it complement\\/}  of $S$, denoted by $S^c$, is the set of elements in the universal set that are not in $S$. \\item % (f)  The {\\it union\\/} of $S$ and $T$, denoted by $S\\cup T$, is the set of elements in at least one of $S$ and $T$ (Figure~\\ref{figure:1.3.1}\\part{b}). \\item % (g) The {\\it intersection\\/}  of $S$ and $T$, denoted by $S\\,\\cap\\, T$, is the set of elements in both $S$ and $T$ (Figure~\\ref{figure:1.3.1}\\part{c}). If $S\\cap T=\\emptyset$ (the empty set), then $S$ and $T$ are  {\\it disjoint sets\\/} (Figure~\\ref{figure:1.3.1}\\part{d}). \\item % (h)  A set with only one member $x_0$ is a {\\it singleton set\\/},  denoted by $\\{x_0\\}$. \\end{alist}"}
{"_id": "300", "title": "", "text": "If $x_0$ is a real number and $\\epsilon>0$, then the open interval $(x_0-\\epsilon, x_0+\\epsilon)$ is an {\\it $\\epsilon$-neighborhood\\/} of $x_0$. If a set $S$ contains an $\\epsilon$-neighborhood of $x_0$, then $S$ is a {\\it neighborhood\\/} of $x_0$, and $x_0$ is an {\\it interior point\\/} of $S$ (Figure~\\ref{figure:1.3.2}). The set of interior points of $S$ is the {\\it interior\\/} of $S$, denoted by $S^0$. If every point of $S$ is an interior point (that is, $S^0=S$), then $S$ is {\\it open\\/}.  A set $S$ is \\emph{closed} if $S^c$ is open."}
{"_id": "301", "title": "", "text": "R}$. Then \\begin{alist} \\item % (a)  $x_0$ is a {\\it limit point\\/} of $S$ if every deleted neighborhood of $x_0$ contains a point of~$S$. \\item % (b) $x_0$ is a {\\it boundary point\\/} of $S$ if every neighborhood of $x_0$ contains at least one point in $S$ and one not in $S$. The set of boundary points of $S$ is the {\\it boundary\\/} of $S$, denoted by $\\partial S$. The {\\it closure\\/} of $S$, denoted by $\\overline{S}$, is $\\overline{S}=S\\cup \\partial S$. \\item % (c) $x_0$ is an \\emph{isolated point} of $S$ if $x_0\\in S$  and there is a neighborhood of $x_0$ that contains no other point of $S$. \\item % (d) $x_0$ is \\emph{exterior} to $S$ if $x_0$ is in the interior of $S^c$. The collection of such points is the {\\it exterior\\/} of $S$. \\end{alist}"}
{"_id": "302", "title": "", "text": "If  $D_f\\cap D_g\\ne \\emptyset,$  then $f+g,$ $f-g,$ and $fg$ are defined on $D_f\\cap D_g$ by \\begin{eqnarray*} (f+g)(x)\\ar= f(x)+g(x),\\\\ (f-g)(x)\\ar= f(x)-g(x),\\\\ \\noalign{\\hbox{and}} (fg)(x)\\ar= f(x)g(x). \\end{eqnarray*} The quotient $f/g$ is defined by $$ \\left(\\frac{f}{ g}\\right) (x)=\\frac{f(x)}{ g(x)} $$ for $x$ in $D_f\\cap D_g$ such that $g(x)\\ne0.$"}
{"_id": "303", "title": "", "text": "We say that $f(x)$ {\\it approaches the limit $L$ as $x$ approaches\\/} $x_0$, and write $$ \\lim_{x\\to x_0} f(x)=L, $$ if $f$ is defined on some deleted neighborhood of $x_0$ and, for every $\\epsilon>0$, there is a $\\delta>0$ such that \\begin{equation}\\label{eq:2.1.4} |f(x)-L|<\\epsilon \\end{equation} if \\begin{equation}\\label{eq:2.1.5} 0<|x-x_0|<\\delta. \\end{equation} Figure~\\ref{figure:2.1.1}   depicts the graph of a function for which $\\lim_{x \\to x_0}f(x)$ exists."}
{"_id": "304", "title": "", "text": "\\begin{alist} \\item % (a) We say that $f(x)$ {\\it approaches the left-hand limit $L$ as $x$ approaches $x_0$ from the left\\/}, and write $$ \\lim_{x\\to x_0-} f(x)=L, $$ if $f$ is defined on some open interval $(a,x_0)$ and, for each $\\epsilon>0$, there is a $\\delta>0$ such that $$ |f(x)-L|<\\epsilon\\mbox{\\quad if \\quad} x_0-\\delta<x<x_0. $$ \\newpage \\item % (b) We say that $f(x)$ {\\it approaches the right-hand limit $L$ as $x$ approaches $x_0$ from the right\\/}, and write $$ \\lim_{x\\to x_0+} f(x)=L, $$ if $f$ is defined on some open interval $(x_0,b)$ and, for each $\\epsilon>0$, there is a $\\delta>0$ such that $$ |f(x)-L|<\\epsilon\\mbox{\\quad if \\quad} x_0<x<x_0+\\delta. \\eqno{\\bbox} $$ \\end{alist}"}
{"_id": "305", "title": "", "text": "We say that $f(x)$ {\\it approaches the limit $L$ as $x$ approaches\\/} $\\infty$, and write $$ \\lim_{x\\to\\infty} f(x)=L, $$ if $f$ is defined on an interval $(a,\\infty)$ and, for each $\\epsilon>0$, there is a number $\\beta$ such that $$ |f(x)-L|<\\epsilon\\quad\\mbox{\\quad if \\quad} x>\\beta. \\eqno{\\bbox} $$"}
{"_id": "306", "title": "", "text": "We say that $f(x)$ {\\it approaches $\\infty$ as $x$ approaches $x_0$ from the left\\/}, and write $$ \\lim_{x\\to x_0-} f(x)=\\infty\\mbox{\\quad or \\quad} f(x_0-)=\\infty, $$ if $f$ is defined on an interval $(a,x_0)$ and, for each real number $M$, there is a $\\delta>0$ such that $$ f(x)>M\\mbox{\\quad if \\quad} x_0-\\delta<x<x_0. $$"}
{"_id": "307", "title": "", "text": "Suppose that  $f$  is bounded on $[a,x_0)$, where $x_0$ may be finite or $\\infty$. For $a\\le x<x_0$, define \\begin{eqnarray*} S_f(x;x_0)\\ar= \\sup_{x\\le t<x_0}f(t)\\\\\\arraytext{and}\\\\ I_f(x;x_0)\\ar= \\inf_{x\\le t<x_0}f(t). \\end{eqnarray*} Then the  {\\it left limit superior of $f$ at $x_0$} is defined to be $$ \\limsup_{x\\to x_0-} f(x)=\\lim_{x\\to x_0-}S_f(x;x_0), $$ and the  {\\it left limit inferior of $f$ at $x_0$} is defined to be $$ \\liminf_{x\\to x_0-}f(x)=\\lim_{x\\to x_0-}I_f(x;x_0). $$ (If $x_0=\\infty$,  we define $x_0-=\\infty$.)"}
{"_id": "308", "title": "", "text": "\\vspace{6pt} \\begin{alist} \\item % (a) We say that $f$ is {\\it continuous at $x_0$\\/} if $f$ is defined on an open interval $(a,b)$ containing $x_0$ and $\\lim_{x\\to x_0}f(x)=f(x_0)$. \\item % (b) We say that $f$ is {\\it continuous from the left at $x_0$\\/} if $f$ is defined on an open interval $(a,x_0)$ and $f(x_0-)=f(x_0)$. \\item % (c) We say that $f$ is {\\it continuous from the right at $x_0$\\/} if $f$ is defined on an open interval $(x_0,b)$ and $f(x_0+)=f(x_0)$. \\bbox \\end{alist}"}
{"_id": "309", "title": "", "text": "A function $f$ is {\\it continuous on an open interval\\/} $(a,b)$ if it is continuous at every point in $(a,b)$. If, in addition, \\begin{eqnarray} f(b-)=f(b)\\label{eq:2.2.2}\\\\ \\arraytext{or}\\nonumber\\\\ f(a+)=f(a)\\label{eq:2.2.3} \\end{eqnarray} \\newpage \\noindent  then $f$ is {\\it continuous on $(a,b]$ or $[a,b)$\\/}, respectively. If $f$ is continuous on $(a,b)$ and \\eqref{eq:2.2.2} and \\eqref{eq:2.2.3} both hold, then {\\it $f$ is continuous on $[a,b]$\\/}. More generally, if $S$ is a subset of $D_f$ consisting of finitely or infinitely many disjoint intervals, then $f$ is {\\it continuous on $S$\\/} if $f$ is continuous on every interval in $S$. (Henceforth, in connection with functions of one variable, whenever we say ``$f$ is continuous on $S$'' we mean that $S$ is a set of this kind.)"}
{"_id": "310", "title": "", "text": "A function $f$ is {\\it piecewise continuous\\/} on $[a,b]$ if \\begin{alist} \\item % (a)  $f(x_0+)$ exists for all $x_0$ in $[a,b)$; \\item % (b)  $f(x_0-)$ exists for all $x_0$ in $(a,b]$; \\item % (c)  $f(x_0+)=f(x_0-)=f(x_0)$ for all but finitely many points $x_0$ in $(a,b)$. \\end{alist} If \\part{c} fails to hold at some $x_0$ in $(a,b)$, $f$ has a {\\it jump discontinuity at $x_0$\\/}. Also, $f$ has a {\\it jump discontinuity at $a$\\/} if $f(a+)\\ne f(a)$ or {\\it at\\/} $b$ if $f(b-)\\ne f(b)$."}
{"_id": "311", "title": "", "text": "Suppose that $f$ and $g$ are functions with domains $D_f$ and $D_g$. If $D_g$ has a nonempty subset $T$ such that  $g(x)\\in D_f$ whenever $x\\in T$, then the {\\it composite function\\/} $f\\circ g$ is defined on $T$ by $$ (f\\circ g)(x)=f(g(x)). $$"}
{"_id": "312", "title": "", "text": "A function $f$ is  {\\it uniformly continuous\\/} on a subset $S$ of its domain if, for every $\\epsilon >0$, there is a $\\delta>0$ such that $$ |f(x)-f(x')|<\\epsilon\\mbox{\\ whenever }\\ |x-x'|<\\delta \\mbox{\\ and }\\ x,x'\\in S. \\eqno{\\bbox} $$"}
{"_id": "313", "title": "", "text": "A function $f$ is {\\it differentiable\\/} at an interior point $x_0$ of its domain if the difference quotient $$ \\frac{f(x)-f(x_0)}{ x-x_0},\\quad x\\ne x_0, $$ approaches a limit as $x$ approaches $x_0$, in which case the limit is called the {\\it derivative of $f$ at $x_0$\\/}, and is denoted by $f'(x_0)$; thus, \\begin{equation}\\label{eq:2.3.1} f'(x_0)=\\lim_{x\\to x_0}\\frac{f(x)-f(x_0)}{ x-x_0}. \\end{equation} It is sometimes convenient to let $x=x_0+h$ and  write \\eqref{eq:2.3.1} as $$ f'(x_0)=\\lim_{h\\to 0}\\frac{f(x_0+h)-f(x_0)}{ h}. \\eqno{\\bbox} $$"}
{"_id": "314", "title": "", "text": "\\begin{alist} \\item % (a) We say that $f$ is {\\it differentiable on the closed interval\\/} $[a,b]$ if $f$ is differentiable on the open interval $(a,b)$ and $f_+'(a)$ and $f_-'(b)$  both exist. \\item % (b) We say that $f$ is {\\it continuously  differentiable on\\/} $[a,b]$ if $f$ is differentiable on  $[a,b]$, $f'$ is continuous on $(a,b)$, $f_+'(a)=f'(a+)$, and $f_-'(b)=f'(b-)$. \\end{alist}"}
{"_id": "315", "title": "", "text": "Let $f$  be defined on $[a,b]$. We say that $f$ is {\\it Riemann integrable  on\\/} $[a,b]$ if there is a number $L$ with the following property: For every $\\epsilon>0$, there is a $\\delta>0$ such that $$ \\left|\\sigma-L \\right|<\\epsilon $$ if $\\sigma$ is any Riemann sum of $f$ over a partition $P$ of $[a,b]$ such that $\\|P\\|<\\delta$. In this case, we say that $L$ is {\\it the Riemann integral of $f$ over\\/} $[a,b]$, and write $$ \\int_a^b f(x)\\,dx=L. $$"}
{"_id": "316", "title": "", "text": "If $f$ is bounded on $[a,b]$ and $P=\\{x_0,x_1, \\dots,x_n\\}$ is a partition of $[a,b]$, let \\begin{eqnarray*} M_j\\ar=\\sup_{x_{j-1}\\le x\\le x_j}f(x)\\\\ \\arraytext{and}\\\\ m_j\\ar=\\inf_{x_{j-1}\\le x\\le x_j}f(x). \\end{eqnarray*} The {\\it upper sum of $f$ over $P$\\/}  is $$ S(P)=\\sum_{j=1}^n M_j(x_j-x_{j-1}), $$ and the {\\it upper integral of $f$ over\\/}, $[a,b]$, denoted by $$ \\overline{\\int_a^b} f(x)\\,dx, $$ is  the infimum of all upper sums.  The {\\it lower sum of $f$ over $P$\\/} is $$ s(P)=\\sum_{j=1}^n m_j(x_j-x_{j-1}), $$ and the {\\it lower integral of $f$ over\\/} $[a,b]$, denoted by $$ \\underline{\\int_a^b}f(x)\\,dx, $$ is the supremum of all lower sums. \\bbox"}
{"_id": "317", "title": "", "text": "Let $f$ and $g$ be defined on $[a,b]$. We say that $f$ is {\\it Riemann}--\\href{http://www-history.mcs.st-and.ac.uk/Mathematicians/Stieltjes.html} {\\it Stieltjes} {\\it integrable with respect to $g$ on\\/} $[a,b]$ if there is a number $L$ with the following property: For every $\\epsilon>0$, there is a $\\delta>0$ such that \\begin{equation} \\label{eq:3.1.15} \\left|\\sum_{j=1}^n f(c_j)\\left[g(x_j)-g(x_{j-1})\\right]-L \\right|< \\epsilon, \\end{equation} provided only that $P=\\{x_0,x_1, \\dots,x_n\\}$ is a partition of $[a,b]$ such that $\\|P\\|<\\delta$ and $$ x_{j-1}\\le c_j\\le x_j,\\quad j=1,2, \\dots,n. $$ In this case, we say that $L$ is {\\it the Riemann--Stieltjes integral of $f$ with respect to $g$ over\\/} $[a,b]$, and write $$ \\int_a^b f(x)\\,dg (x)=L. $$ The sum $$ \\sum_{j=1}^n f(c_j)\\left[g(x_j)-g(x_{j-1})\\right] $$ in \\eqref{eq:3.1.15} is  {\\it a Riemann--Stieltjes sum of $f$ with respect to $g$ over the partition~$P$\\/}."}
{"_id": "318", "title": "", "text": "If $f$ is locally integrable on $[a,b)$, we define \\begin{equation}\\label{eq:3.4.1} \\int_a^b f(x)\\,dx= \\lim_{c\\to b-}\\int_a^c f(x)\\,dx \\end{equation} if the limit exists (finite). To include the case where $b=\\infty$, we adopt the convention that $\\infty-=\\infty$. \\bbox"}
{"_id": "319", "title": "", "text": "If $f$ is locally integrable on $(a,b]$, we define $$ \\int_a^b f(x)\\,dx=\\lim_{c\\to a+}\\int_c^b f(x)\\,dx $$ provided that the limit exists (finite).  To include the case where $a=-\\infty$, we adopt the convention that $-\\infty+=-\\infty$."}
{"_id": "320", "title": "", "text": "If $f$ is locally integrable on $(a,b),$ we define $$ \\int_a^b f(x)\\,dx=\\int_a^\\alpha f(x)\\,dx+\\int_\\alpha^b f(x)\\,dx, $$ where $a<\\alpha<b$, provided that both improper integrals on the right exist (finite). \\bbox"}
{"_id": "321", "title": "", "text": "We say that $f$ is {\\it absolutely integrable on\\/} $[a,b)$ if $f$ is locally integrable on $[a,b)$ and $\\int_a^b|f(x)|\\,dx<\\infty$.  In this case we also say that $\\int_a^b f(x)\\,dx$ {\\it converges absolutely\\/} or {\\it is absolutely convergent\\/}."}
{"_id": "322", "title": "", "text": "If $f$ is bounded on $[a,b]$, the {\\it oscillation of $f$ on\\/} $[a,b]$ is defined by $$ W_f[a,b]=\\sup_{a\\le x,x'\\le b}|f(x)-f(x')|, $$ which can also be written as $$ W_f[a,b]=\\sup_{a\\le x\\le b}f(x)-\\inf_{a\\le x\\le b}f(x) $$ \\newpage \\noindent ( Exercise~\\ref{exer:3.5.1}). If $a<x<b$,  the {\\it oscillation of $f$ at $x$\\/} is defined by $$ w_f(x)=\\lim_{h\\to0+}W_f(x-h,x+h). $$ The corresponding definitions for $x=a$ and $x=b$ are $$ w_f(a)=\\lim_{h\\to0+}W_f(a,a+h)\\mbox{\\quad and \\quad} w_f(b)=\\lim_{h\\to0+}W_f(b-h,b). \\eqno{\\bbox} $$"}
{"_id": "323", "title": "", "text": "A subset $S$ of the real line is {\\it of Lebesgue measure zero\\/} if for every $\\epsilon>0$ there is a finite or infinite sequence of open intervals $I_1$, $I_2$, \\dots\\ such that \\begin{equation} \\label{eq:3.5.8} S\\subset\\bigcup_j I_j \\end{equation} and \\begin{equation} \\label{eq:3.5.9} \\sum_{j=1}^n L(I_j)<\\epsilon,\\quad n\\ge1. \\end{equation}"}
{"_id": "324", "title": "", "text": "A sequence $\\{s_n\\}$ {\\it converges to a limit $s$\\/} if for every  $\\epsilon>0$ there is an integer $N$ such that \\begin{equation}\\label{eq:4.1.2} |s_n-s|<\\epsilon\\mbox{\\quad if\\quad} n\\ge N. \\end{equation} In this case we say that $\\{s_n\\}$ is {\\it convergent\\/} and write $$ \\lim_{n\\to\\infty}s_n=s. $$ A sequence that does not converge {\\it diverges\\/}, or is {\\it divergent\\/} \\bbox"}
{"_id": "325", "title": "", "text": "A sequence $\\{s_n\\}$ is  {\\it bounded above\\/}  if there is a real number $b$ such that $$ s_n\\le b\\mbox{\\quad for all $n$}, $$ {\\it bounded below\\/} if there is a real number $a$ such that $$ s_n\\ge a\\mbox{\\quad for all $n$}, $$ or {\\it bounded\\/} if there is a real number $r$ such that $$ |s_n|\\le r\\mbox{\\quad for all $n$}. $$"}
{"_id": "326", "title": "", "text": "A sequence $\\{s_n\\}$ is {\\it nondecreasing\\/} if $s_n\\ge s_{n-1}$ for all $n$, or {\\it nonincreasing\\/} if $s_n\\le s_{n-1}$ for all $n.$ A {\\it monotonic sequence\\/} is a sequence that is either nonincreasing or nondecreasing. If $s_n>s_{n-1}$ for all $n$, then $\\{s_n\\}$ is  {\\it increasing\\/}, while if $s_n<s_{n-1}$ for all $n$, $\\{s_n\\}$ is {\\it decreasing\\/}."}
{"_id": "327", "title": "", "text": "The numbers $\\overline{s}$ and $\\underline{s}$ defined in Theorem~\\ref{thmtype:4.1.9} are called the {\\it limit superior\\/} and {\\it limit inferior\\/}, respectively, of $\\{s_n\\}$, and denoted by $$ \\overline{s}=\\limsup_{n\\to\\infty} s_n\\mbox{\\quad and\\quad} \\underline{s}=\\liminf_{n\\to\\infty} s_n. $$ We also define \\begin{eqnarray*} \\limsup_{n\\to\\infty}s_n\\ar=\\phantom{-}\\infty\\quad\\mbox{if $\\{s_n\\}$ is not bounded above},\\\\ \\limsup_{n\\to\\infty}s_n\\ar=-\\infty\\quad\\mbox{if }\\lim_{n\\to \\infty}s_n=-\\infty,\\\\ \\liminf_{n\\to\\infty} s_n\\ar=-\\infty\\quad\\mbox{if $\\{s_n\\}$ is not bounded below},\\\\ \\arraytext{and}[2\\jot] \\liminf_{n\\to\\infty} s_n\\ar=\\phantom{-}\\infty\\quad\\mbox{if } \\lim_{n\\to\\infty} s_n=\\infty. \\end{eqnarray*}"}
{"_id": "328", "title": "", "text": "A sequence $\\{t_k\\}$ is  a {\\it subsequence\\/} of a sequence $\\{s_n\\}$ if $$ t_k=s_{n_k},\\quad k\\ge0, $$ where $\\{n_k\\}$ is an increasing infinite sequence of integers in the domain of $\\{s_n\\}$.  We denote the subsequence $\\{t_k\\}$ by $\\{s_{n_k}\\}$. \\bbox"}
{"_id": "329", "title": "", "text": "If $\\{a_n\\}_k^\\infty$ is an infinite sequence of real numbers, the symbol $$ \\sum_{n=k}^\\infty a_n $$ \\newpage \\noindent is an {\\it infinite series\\/}, and $a_n$ is the {\\it $n$th term\\/} of the series. We say that $\\sum_{n=k}^\\infty a_n$ {\\it converges to the \\enlargethispage{\\baselineskip} sum $A$\\/},  and write $$ \\sum_{n=k}^\\infty a_n=A, $$ if the sequence $\\{A_n\\}_k^\\infty$ defined by $$ A_n=a_k+a_{k+1}+\\cdots+a_n,\\quad n\\ge k, $$ converges to $A$. The finite sum $A_n$ is  the {\\it $n$th partial sum of\\/} $\\sum_{n=k}^\\infty a_n$. If $\\{A_n\\}_k^\\infty$ diverges, we say that $\\sum_{n=k}^\\infty a_n$ {\\it diverges\\/};  in particular, if $\\lim_{n\\to\\infty}A_n =\\infty$ or $-\\infty$, we say that $\\sum_{n=k}^\\infty a_n$ {\\it diverges to $\\infty$ or $-\\infty$\\/}, and write $$ \\sum_{n=k}^\\infty a_n=\\infty\\mbox{\\quad or\\quad}\\sum_{n=k}^\\infty a_n= -\\infty. $$ A divergent infinite series that does not diverge to $\\pm\\infty$ is said to {\\it oscillate\\/}, or {\\it be oscillatory\\/}. \\bbox"}
{"_id": "330", "title": "", "text": "{\\it converges absolutely\\/}, or is {\\it  absolutely convergent\\/}$,$ if $\\sum|a_n|<\\infty.$"}
{"_id": "331", "title": "", "text": "The {\\it Cauchy product of $\\sum_{n=0\\/}^\\infty a_n$ and\\/} $\\sum_{n=0}^\\infty b_n$ is $\\sum_{n=0}^\\infty c_n$, where \\begin{equation}\\label{eq:4.3.38} c_n=a_0b_n+a_1b_{n-1}+\\cdots+a_{n-1}b_1+a_nb_0. \\end{equation} Thus, $c_n$ is the sum of all products $a_ib_j$, where $i\\ge0$, $j\\ge0$, and $i+j=n$; thus, \\begin{equation} \\label{eq:4.3.39} c_n=\\sum_{r=0}^n a_rb_{n-r}=\\sum_{r=0}^n b_ra_{n-r}. \\end{equation}"}
{"_id": "332", "title": "", "text": "Suppose that  $\\{F_n\\}$ is a sequence of functions on $D$ and the sequence of values $\\{F_n(x)\\}$ converges for each $x$ in some subset $S$ of $D$. Then we say that $\\{F_n\\}$ {\\it converges pointwise on $S$ to the limit function\\/} $F$, defined by $$ F(x)=\\lim_{n\\to\\infty}F_n(x),\\quad x\\in S. $$"}
{"_id": "333", "title": "", "text": "A sequence $\\{F_n\\}$ of functions defined on a set $S$ {\\it converges uniformly to the limit function $F$ on\\/} $S$ if $$ \\lim_{n\\to\\infty} || F_n-F\\|_S=0. $$ Thus, $\\{F_n\\}$ converges uniformly to $F$ on $S$ if for each $\\epsilon> 0$ there is an integer $N$ such that \\begin{equation} \\label{eq:4.4.1} \\|F_n-F\\|_S<\\epsilon\\mbox{\\quad if\\quad} n\\ge N. \\end{equation}"}
{"_id": "334", "title": "", "text": "If $\\{f_j\\}^\\infty_k$ is a sequence of real-valued functions defined on a set $D$ of reals, then $\\sum_{j=k}^\\infty  f_j$ is  an {\\it infinite series\\/} (or simply a {\\it series\\/}) of functions on $D$. The {\\it partial sums of\\/}, $\\sum_{j=k}^\\infty f_j$ are defined by $$ F_n=\\sum^n_{j=k} f_j,\\quad n\\ge k. $$ If $\\{F_n\\}^\\infty_k$ converges pointwise to a function $F$ on a subset $S$ of $D$, we say that $\\sum_{j=k}^\\infty f_j$ {\\it converges pointwise to the sum $F$ on\\/} $S$, and write $$ F=\\sum_{j=k}^\\infty f_j,\\quad x\\in S. $$ \\newpage \\noindent If $\\{F_n\\}$ converges uniformly to $F$ on $S$, we say that $\\sum_{j=k}^\\infty f_j$ {\\it converges uniformly to $F$ on~$S$\\/}."}
{"_id": "335", "title": "", "text": "An infinite series of the form \\begin{equation}\\label{eq:4.5.1} \\sum^\\infty_{n=0} a_n(x-x_0)^n, \\end{equation} where $x_0$ and $a_0$, $a_1$, \\dots, are constants, is called a {\\it power series in $x-x_0$\\/}. \\bbox"}
{"_id": "336", "title": "", "text": "The {\\it vector sum\\/} of $$ \\mathbf{X}=(x_1,x_2, \\dots,x_n)\\mbox{\\quad and\\quad}\\mathbf{Y}= (y_1,y_2, \\dots,y_n) $$ is \\begin{equation}\\label{eq:5.1.1} \\mathbf{X}+\\mathbf{Y}=(x_1+y_1,x_2+y_2, \\dots,x_n+y_n). \\end{equation} If $a$ is a real number, the {\\it scalar multiple of $\\mathbf{X\\/}$ by\\/} $a$ is \\begin{equation}\\label{eq:5.1.2} a\\mathbf{X}=(ax_1,ax_2, \\dots,ax_n). \\end{equation}"}
{"_id": "337", "title": "", "text": "The {\\it length\\/} of the vector $\\mathbf{X}=(x_1,x_2, \\dots, x_n)$ is $$ |\\mathbf{X}|=(x^2_1+x^2_2+\\cdots+x^2_n)^{1/2}. $$ The {\\it distance between points $\\mathbf{X\\/}$ and\\/} $\\mathbf{Y}$ is $|\\mathbf{X}-\\mathbf{Y}|$; in particular, $|\\mathbf{X}|$ is the distance between $\\mathbf{X}$ and the origin. If $|\\mathbf{X}|=1$, then $\\mathbf{X}$  is a {\\it unit vector\\/}. \\bbox"}
{"_id": "338", "title": "", "text": "The {\\it inner product\\/} $\\mathbf{X}\\cdot \\mathbf{Y}$ of $\\mathbf{X}=(x_1,x_2, \\dots,x_n)$ and $\\mathbf{Y}= (y_1,y_2, \\dots,y_n)$ is $$ \\mathbf{X}\\cdot\\mathbf{Y}=x_1y_1+x_2y_2+\\cdots+x_ny_n. $$"}
{"_id": "339", "title": "", "text": "$\\mathbf{U}$ are in $\\R^n$ and $\\mathbf{U}\\ne\\mathbf{0}$. Then {\\it the line through $\\mathbf{X}_0$ in the direction of\\/} $\\mathbf{U}$ is the set of all points in $\\R^n$ of the form $$ \\mathbf{X}=\\mathbf{X}_0+t\\mathbf{U},\\quad -\\infty<t<\\infty. $$ A set of points of the form $$ \\mathbf{X}=\\mathbf{X}_0+t\\mathbf{U},\\quad t_1\\le t\\le t_2, $$ is called a {\\it line segment\\/}. In particular, the line segment from $\\mathbf{X}_0$ to $\\mathbf{X}_1$ is the set of points of the form $$ \\mathbf{X}=\\mathbf{X}_0+t(\\mathbf{X}_1-\\mathbf{X}_0)=t\\mathbf{X}_1+(1-t)\\mathbf{X}_0, \\quad 0\\le t\\le1. $$"}
{"_id": "340", "title": "", "text": "If $\\epsilon>0$, the {\\it $\\epsilon$-neighborhood of a point\\/} $\\mathbf{X}_{0}$ in $\\R^n$ is the set $$ N_\\epsilon(\\mathbf{X}_0)|=\\set{\\mathbf{X}}{|\\mathbf{X}-\\mathbf{X}_0|<\\epsilon}. \\eqno{\\bbox} $$"}
{"_id": "341", "title": "", "text": "A sequence of points $\\{\\mathbf{X}_r\\}$ in $\\R^n$ {\\it converges to the limit\\/} $\\overline{\\mathbf{X}}$ if $$ \\lim_{r\\to\\infty} |\\mathbf{X}_r-\\overline{\\mathbf{X}}|=0. $$ In this case we write $$ \\lim_{r\\to\\infty}\\mathbf{X}_r=\\overline{\\mathbf{X}}. \\eqno{\\bbox} $$"}
{"_id": "342", "title": "", "text": "If $S$ is  a nonempty subset of $\\R^n$, then $$ d(S)=\\sup\\set{|\\mathbf{X}-\\mathbf{Y}|}{\\mathbf{X},\\mathbf{Y}\\in S} $$ is the {\\it diameter\\/} of $S$. If $d(S)<\\infty,$ $S$ is  {\\it bounded\\/}$;$ if $d(S)=\\infty$, $S$ is {\\it unbounded\\/}."}
{"_id": "343", "title": "", "text": "A subset $S$ of $\\R^n$ is  {\\it connected\\/} if it is impossible to represent $S$ as the union of two disjoint nonempty sets such that neither contains a limit point of the other; that is, if $S$ cannot be expressed as $S=A\\cup B$, where \\begin{equation}\\label{eq:5.1.16} A\\ne\\emptyset,\\quad B\\ne\\emptyset,\\quad\\overline{A}\\cap B= \\emptyset,\\mbox{\\quad and\\quad} A\\cap\\overline{B}=\\emptyset. \\end{equation} If $S$ can be expressed in this way, then $S$ is {\\it disconnected\\/}."}
{"_id": "344", "title": "", "text": "A {\\it region\\/} $S$ in $\\R^n$ is the union of an open connected set with some, all, or none of its boundary; thus, $S^0$ is connected, and every point of $S$ is a limit point of $S^0$."}
{"_id": "345", "title": "", "text": "We say that $f(\\mathbf{X})$ {\\it approaches the limit $L$ as $\\mathbf{X\\/}$ approaches\\/} $\\mathbf{X}_0$ and write $$ \\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=L $$ if $\\mathbf{X}_0$ is a limit point of $D_f$ and, for every $\\epsilon>0$, there is a $\\delta>0$ such that $$ |f(\\mathbf{X})-L|<\\epsilon $$ for all $\\mathbf{X}$ in $D_f$ such that $$ 0<|\\mathbf{X}-\\mathbf{X}_0|<\\delta. $$"}
{"_id": "346", "title": "", "text": "We say that $f(\\mathbf{X})$ {\\it approaches $\\infty$ as $\\mathbf{X\\/}$ approaches $\\mathbf{X}_0$\\/} and write $$ \\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=\\infty $$ if $\\mathbf{X}_0$ is a limit point of $D_f$ and, for every real number $M$, there is a $\\delta>0$ such that $$ f(\\mathbf{X})>M\\mbox{\\quad whenever\\quad} 0<|\\mathbf{X}-\\mathbf{X}_0|<\\delta \\mbox{\\quad and\\quad}\\mathbf{X}\\in D_f. $$ We say that \\begin{eqnarray*} \\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})\\ar=-\\infty\\\\ \\arraytext{if}\\\\ \\lim_{{\\mathbf{X}}\\to\\mathbf{X}_0} (-f)(\\mathbf{X})\\ar=\\infty. \\end{eqnarray*}"}
{"_id": "347", "title": "", "text": "If $D_f$ is unbounded$,$ we say that $$ \\lim_{|\\mathbf{X}|\\to\\infty} f(\\mathbf{X})=L\\mbox{\\quad (finite)\\quad} $$ if for every $\\epsilon>0$, there is a number $R$ such that $$ |f(\\mathbf{X})-L|<\\epsilon\\mbox{\\quad whenever\\quad}\\ |\\mathbf{X}|\\ge R \\mbox{\\quad and\\quad}\\mathbf{X}\\in D_f. $$"}
{"_id": "348", "title": "", "text": "If $\\mathbf{X}_0$ is in $D_f$ and is a limit point of $D_f$, then we say that $f$ is {\\it continuous at  $\\mathbf{X\\/}_0$\\/}  if $$ \\lim_{\\mathbf{X}\\to\\mathbf{X}_0} f(\\mathbf{X})=f(\\mathbf{X}_0). \\eqno{\\bbox} $$"}
{"_id": "349", "title": "", "text": "Let $\\boldsymbol{\\Phi}$ be a unit vector and $\\mathbf{X}$ a point in $\\R^n$.  {\\it The directional derivative of $f$ at $\\mathbf{X}$ in the direction of\\/} $\\boldsymbol{\\Phi}$ is defined by $$ \\frac{\\partial f(\\mathbf{X})}{\\partial\\boldsymbol{\\Phi}}=\\lim_{t\\to 0}\\frac {f(\\mathbf{X}+ t\\boldsymbol{\\Phi})-f(\\mathbf{X})}{ t} $$ if the limit exists. That is, $\\partial f(\\mathbf{X})/\\partial\\boldsymbol{\\Phi}$ is the ordinary derivative of the function $$ h(t)=f(\\mathbf{X}+t\\boldsymbol{\\Phi}) $$ at $t=0$, if $h'(0)$ exists."}
{"_id": "350", "title": "", "text": "A function $f$ is {\\it differentiable\\/} at $$  \\mathbf{X}_0=(x_{10},x_{20}, \\dots,x_{n0})) $$ if $\\mathbf{X}_0\\in D_f^0$ and there are  constants $m_1$, $m_2$, \\dots$,$ $m_n$ such that \\begin{equation}\\label{eq:5.3.16} \\lim_{\\mathbf{X}\\to\\mathbf{X}_0} \\frac{f(\\mathbf{X})-f(\\mathbf{X}_0)- \\dst{\\sum^n_{i=1}}\\, m_i (x_i-x_{i0})}{ |\\mathbf{X}-\\mathbf{X}_0|}=0. \\end{equation}"}
{"_id": "351", "title": "", "text": "A vector-valued function  $\\mathbf{G}=(g_1,g_2, \\dots,g_n)$ is {\\it differentiable\\/} at $$ \\mathbf{U}_0=(u_{10},u_{20}, \\dots,u_{m0}) $$  if its component functions $g_1$, $g_2$, \\dots, $g_n$ are differentiable at $\\mathbf{U}_0$. \\bbox"}
{"_id": "352", "title": "", "text": "Suppose that  $r\\ge1$ and all partial derivatives of $f$ of order $\\le r-1$ are differentiable in a neighborhood of $\\mathbf{X}_0$. Then the $r$th {\\it differential of $f$ at\\/} $\\mathbf{X}_0$, denoted by $d^{(r)}_{\\mathbf{X}_0}f$, is defined by \\begin{equation} \\label{eq:5.4.23} d^{(r)}_{\\mathbf{X}_0}f=\\sum_{i_1,i_2, \\dots,i_r=1}^n \\frac{\\partial^rf(\\mathbf{X}_0) }{\\partial x_{i_r}\\partial x_{i_{r-1}}\\cdots\\partial x_{i_1}} dx_{i_1}dx_{i_2}\\cdots dx_{i_r}, \\end{equation} where $dx_1$, $dx_2$, \\dots, $dx_n$ are the differentials introduced in Section~5.3; that is, $dx_i$ is the function whose value at a point in $\\R^n$ is the $i$th coordinate of the point. For convenience, we define $$ (d^{(0)}_{\\mathbf{X}_0}f)=f(\\mathbf{X}_0). $$ Notice that $d^{(1)}_{\\mathbf{X}_0}f=d_{\\mathbf{X}_0}f$. \\bbox"}
{"_id": "353", "title": "", "text": "A transformation $\\mathbf{L}: \\R^n \\to \\R^m$ defined on all of $\\R^n$ is  {\\it linear\\/} if $$ \\mathbf{L}(\\mathbf{X}+\\mathbf{Y})=\\mathbf{L}(\\mathbf{X})+\\mathbf{L}(\\mathbf{Y}) $$ for all $\\mathbf{X}$ and $\\mathbf{Y}$ in $\\R^n$ and $$ \\mathbf{L}(a\\mathbf{X})=a\\mathbf{L}(\\mathbf{X}) $$ for all $\\mathbf{X}$ in $\\R^n$ and real numbers $a$."}
{"_id": "354", "title": "", "text": "\\begin{alist} \\item % (a)  If $c$ is a real number and $\\mathbf{A}=[a_{ij}]$ is an $m\\times n$ matrix, then $c\\mathbf{A}$ is the $m\\times n$ matrix defined by $$ c\\mathbf{A}=[ca_{ij}]; $$ that is, $c\\mathbf{A}$ is obtained by multiplying every entry of $\\mathbf{A}$ by $c$. \\item % (b) If $\\mathbf{A}=[a_{ij}]$ and $\\mathbf{B}=[b_{ij}]$ are $m\\times n$ matrices, then the {\\it sum\\/}  $\\mathbf{A}+  \\mathbf{B}$  is the $m\\times n$ matrix $$ \\mathbf{A}+\\mathbf{B}=[a_{ij}+b_{ij}]; $$ that is, the sum of two $m\\times n$ matrices is obtained by adding corresponding entries. The sum of two matrices is not defined unless they have the same number of rows and the same number of columns. \\item % (c) If $\\mathbf{A}=[a_{ij}]$ is an $m\\times p$ matrix and $\\mathbf{B}= [b_{ij}]$ is a $p\\times n$ matrix, then the {\\it product\\/} $\\mathbf{C}=\\mathbf{AB}$ is the $m\\times n$ matrix with $$ c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\\cdots+a_{ip}b_{pj}=\\sum^p_{k=1} a_{ik}b_{kj},\\quad 1\\le i\\le m,\\ 1\\le j\\le n. $$ Thus, the $(i,j)$th entry of $\\mathbf{AB}$ is obtained by multiplying each entry in the $i$th row of $\\mathbf{A}$ by the corresponding entry in the $j$th column of $\\mathbf{B}$ and adding the products. This definition requires that $\\mathbf{A}$ have the same number of columns as $\\mathbf{B}$ has rows. Otherwise, $\\mathbf{AB}$ is undefined. \\end{alist}"}
{"_id": "355", "title": "", "text": "The {\\it norm\\/}$,$ $\\|\\mathbf{A}\\|,$ of an $m\\times n$ matrix $\\mathbf{A}=[a_{ij}]$ is the smallest number such that $$ |\\mathbf{AX}|\\le\\|\\mathbf{A}\\|\\,|\\mathbf{X}| $$ for all $\\mathbf{X}$ in $\\R^n.$ \\bbox"}
{"_id": "356", "title": "", "text": "Let $\\mathbf{A}=[a_{ij}]$ be an $n\\times n$ matrix$,$ with $n\\ge2.$ The {\\it cofactor\\/} of an entry $a_{ij}$ is $$ c_{ij}=(-1)^{i+j}\\det(\\mathbf{A}_{ij}), $$ where $\\mathbf{A}_{ij}$ is the $(n-1)\\times(n-1)$ matrix obtained by deleting  the $i$th row and $j$th column of $\\mathbf{A}.$ The {\\it adjoint\\/} of $\\mathbf{A},$ denoted by $\\adj(\\mathbf{A}),$ is the $n\\times n$ matrix whose $(i,j)$th entry is $c_{ji}.$"}
{"_id": "357", "title": "", "text": "A transformation $\\mathbf{F}: \\R^n\\to \\R^n$ is {\\it regular\\/} on an open set $S$ if $\\mathbf{F}$ is one-to-one and continuously differentiable on $S$, and $J\\mathbf{F}(\\mathbf{X})\\ne0$ if $\\mathbf{X}\\in S$. We will also say that $\\mathbf{F}$  is regular on an arbitrary  set $S$ if $\\mathbf{F}$ is regular on an open set containing $S$."}
{"_id": "358", "title": "", "text": "A {\\it coordinate rectangle\\/} $R$ in $\\R^n$ is the Cartesian product of $n$ closed intervals; that is, $$ R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n]. $$ The {\\it content\\/} of $R$ is $$ V(R)=(b_1-a_1)(b_2-a_2)\\cdots (b_n-a_n). $$ The numbers $b_1-a_1$, $b_2-a_2$, \\dots, $b_n-a_n$ are  the {\\it edge lengths\\/} of $R$. If they are equal, then $R$ is a {\\it coordinate cube\\/}.  If $a_r=b_r$ for some $r$, then $V(R)=0$ and we say that $R$ is {\\it degenerate\\/}; otherwise, $R$ is {\\it nondegenerate\\/}.  \\bbox"}
{"_id": "359", "title": "", "text": "Let $f$ be a real-valued function defined on a  rectangle $R$ in $\\R^n$. We say that  $f$ is {\\it Riemann integrable on\\/} $R$  if there is a number $L$ with the following property: For every $\\epsilon>0$, there is a $\\delta>0$ such that $$ \\left|\\sigma-L\\right|<\\epsilon $$ if $\\sigma$ is any Riemann sum of $f$ over a partition ${\\bf P}$ of $R$ such that $\\|{\\bf P}\\|<\\delta$. In this case, we say that  $L$ is the  {\\it Riemann integral of $f$ over\\/} $R$, and write $$ \\int_R f(\\mathbf{X})\\,d\\mathbf{X}=L. \\eqno{\\bbox} $$"}
{"_id": "360", "title": "", "text": "If $f$ is bounded on a rectangle $R$ in $\\R^n$ and ${\\bf P}=\\{R_1,R_2, \\dots,R_k\\}$ is a partition of $R$, let $$ M_j=\\sup_{\\mathbf{X}\\in R_j}f(\\mathbf{X}),\\quad m_j= \\inf_{\\mathbf{X}\\in R_j}f(\\mathbf{X}). $$ The {\\it upper sum\\/} of $f$ over ${\\bf P}$ is $$ S({\\bf P})=\\sum_{j=1}^k M_jV(R_j), $$ and the {\\it upper integral  of $f$ over\\/} $R$, denoted by $$ \\overline{\\int_R}\\,f(\\mathbf{X})\\,d\\mathbf{X}, $$  is the infimum of all  upper sums. The {\\it lower sum of $f$ over\\/} ${\\bf P}$ is $$ s({\\bf P})=\\sum_{j=1}^k m_jV(R_j), $$ and the {\\it lower integral  of $f$ over \\/}$R$, denoted by $$ \\underline{\\int_R}\\, f(\\mathbf{X})\\,d\\mathbf{X}, $$  is the supremum of all  lower sums. \\bbox"}
{"_id": "361", "title": "", "text": "A subset $E$ of $\\R^n$ has zero content if for each $\\epsilon>0$ there is a finite set of rectangles $T_1$, $T_2$, \\dots, $T_m$ such that \\begin{equation}\\label{eq:7.1.24} E\\subset\\bigcup_{j=1}^m T_j \\end{equation} and \\begin{equation}\\label{eq:7.1.25} \\sum_{j=1}^m V(T_j)<\\epsilon. \\end{equation}"}
{"_id": "362", "title": "", "text": "Suppose that $f$ is bounded on a bounded subset of $S$ of $\\R^n$, and let \\begin{equation}\\label{eq:7.1.36} f_S(\\mathbf{X})=\\left\\{\\casespace\\begin{array}{ll} f(\\mathbf{X}),&\\mathbf{X}\\in S,\\\\[2\\jot]  0,&\\mathbf{X}\\not\\in S.\\end{array}\\right. \\end{equation} Let $R$ be a rectangle containing $S$. Then {\\it the integral of $f$ over $S$\\/} is defined to be $$ \\int_S f(\\mathbf{X})\\,d\\mathbf{X}=\\int_R f_S(\\mathbf{X})\\,d\\mathbf{X} $$ if  $\\int_R f_S(\\mathbf{X})\\, d\\mathbf{X}$ exists. \\bbox"}
{"_id": "363", "title": "", "text": "If $S$ is a bounded subset of $\\R^n$ and the integral $\\int_S\\,d\\mathbf{X}$ (with integrand  $f\\equiv1$) exists, we call $\\int_S\\,d\\mathbf{X}$ the {\\it content\\/} (also, {\\it area\\/} if $n=2$ or {\\it volume\\/} if $n=3$) of $S$, and denote it by $V(S)$; thus, $$ V(S)=\\int_S\\,d\\mathbf{X}. $$"}
{"_id": "364", "title": "", "text": "A {\\it differentiable surface\\/} $S$ in $\\R^n\\ (n>1)$ is the image of a compact subset $D$ of $\\R^m$, where $m< n$, under a continuously differentiable transformation $\\mathbf{G}: \\R^m\\to \\R^n$. If $m=1$, $S$ is also called a {\\it differentiable curve\\/}."}
{"_id": "365", "title": "", "text": "If $\\mathbf{A}=[a_{ij}]$ is an $n \\times n$ matrix$,$ then $$ \\max\\set{\\sum_{j=1}^n |a_{ij}|}{1\\le i\\le n} $$ is the {\\it infinity norm of\\/} $A,$  denoted by $\\|A\\|_\\infty$."}
{"_id": "366", "title": "", "text": "A {\\it metric space\\/} is a nonempty set $A$ together with a real-valued function $\\rho$ defined on $A\\times A$ such that  if $u$, $v$, and $w$ are arbitrary members of $A$, then \\begin{alist} \\item % (a) $\\rho(u,v)\\ge 0$, with equality if and only if $u=v$; \\item % (b) $\\rho(u,v)=\\rho(v,u)$; \\item % (c) $\\rho(u,v)\\le\\rho(u,w)+\\rho(w,v)$. \\end{alist} We say that $\\rho$ is a {\\it metric\\/} on $A$. \\bbox"}
{"_id": "367", "title": "", "text": "A {\\it vector space\\/} $A$ is a nonempty set of elements called {\\it vectors\\/} on which two operations,  vector addition and scalar multiplication (multiplication by real numbers) are defined, such that the following assertions are true for all $\\mathbf{U}$, $\\mathbf{V}$, and $\\mathbf{W}$ in $A$ and all real numbers $r$ and $s$:\\\\ \\phantom{1}1.  $\\mathbf{U}+\\mathbf{V}\\in A$;\\\\ \\phantom{1}2. $\\mathbf{U}+\\mathbf{V}=\\mathbf{V}+\\mathbf{U}$;\\\\ \\phantom{1}3.  $\\mathbf{U}+(\\mathbf{V}+\\mathbf{W})=(\\mathbf{U}+\\mathbf{V})+\\mathbf{W}$;\\\\ \\phantom{1}4. There is a vector $\\mathbf{0}$ in $A$ such that $\\mathbf{U}+\\mathbf{0}=\\mathbf{U}$;\\\\ \\phantom{1}5. There is a vector $-\\mathbf{U}$ in $A$ such that $\\mathbf{U}+(-\\mathbf{U})=\\mathbf{0}$;\\\\ \\phantom{1}6. $r\\mathbf{U}\\in A$;\\\\ \\phantom{1}7. $r(\\mathbf{U}+\\mathbf{V})=r\\mathbf{U}+r\\mathbf{V}$;\\\\ \\phantom{1}8. $(r+s)\\mathbf{U}=r\\mathbf{U}+s\\mathbf{U}$;\\\\ \\phantom{1}9. $r(s\\mathbf{U})=(rs)\\mathbf{U}$; \\\\ 10. $1\\mathbf{U}=\\mathbf{U}$. \\bbox"}
{"_id": "368", "title": "", "text": "A {\\it normed vector space\\/}  is a vector space $A$ together with a real-valued function $N$  defined on $A$, such that  if $u$ and $v$ are arbitrary vectors in $A$ and $a$ is a real number, then \\begin{alist} \\item % (a) $N(u)\\ge 0$ with equality if and only if $u=0$; \\item % (b) $N(au)=|a|N(u)$; \\item % (c) $N(u+v)\\le N(u)+N(v)$. \\end{alist} We say that $N$ is a {\\it norm\\/} on $A$, and $(A,N)$ is a {\\it normed vector space\\/}."}
{"_id": "369", "title": "", "text": "If $p\\ge 1$ and $\\mathbf{X}=(x_1,x_2, \\dots,x_n)$, let \\begin{equation} \\label{eq:8.1.3} \\|\\mathbf{X}\\|_p =\\left(\\sum_{i=1}^n|x_i|^p\\right)^{1/p}. \\end{equation} The metric induced on $\\R^n$ by this norm is $$ \\rho_p(\\mathbf{X},\\mathbf{Y}) =\\left(\\sum_{i=1}^n|x_i-y_i|^p\\right)^{1/p}. \\eqno{\\bbox} $$"}
{"_id": "370", "title": "", "text": "If $u_0\\in A$  and $\\epsilon>0$, the set $$ N_\\epsilon(u_0)=\\set{u\\in A}{\\rho(u_0,u)<\\epsilon} $$ is called an {\\it $\\epsilon$-neighborhood\\/} of $u_0$. (Sometimes we call $S_\\epsilon$ the {\\it open ball of radius $\\epsilon$ centered at $u_0$\\/}.) If a subset $S$ of $A$ contains an $\\epsilon$-neighborhood of $u_0$, then $S$ is a {\\it neighborhood\\/} of $u_0$, and $u_0$ is an {\\it interior point\\/} of $S$. The set of interior points of $S$ is the {\\it interior\\/} of $S$, denoted by $S^0$. If every point of $S$ is an interior point (that is, $S^0=S$), then $S$ is {\\it open\\/}. A set $S$ is  {\\it closed\\/} if $S^c$ is open."}
{"_id": "371", "title": "", "text": "Then \\begin{alist} \\item % (a) $u_0$ is a {\\it limit point\\/} of $S$ if every deleted neighborhood of $u_0$ contains a point of~$S$. \\item % (b) $u_0$ is a {\\it boundary point\\/} of $S$ if every neighborhood of $u_0$ contains at least one point in $S$ and one not in $S$. The set of boundary points of $S$ is the {\\it boundary\\/} of $S$, denoted by $\\partial S$. The {\\it closure\\/} of $S$, denoted by $\\overline{S}$, is defined by $\\overline{S}=S\\cup \\partial S$. \\item % (c) $u_0$ is an {\\it isolated point\\/} of $S$ if $u_0\\in S$ and there is a neighborhood of $u_0$ that contains no other point of $S$. \\item % (d) $u_0$ is {\\it exterior } to $S$ if $u_0$ is in the interior of $S^c$. The collection of such points is the {\\it exterior\\/} of $S$. \\bbox \\end{alist}"}
{"_id": "372", "title": "", "text": "A sequence $\\{u_n\\}$ in a metric space $(A,\\rho)$  {\\it converges\\/} to $u\\in A$ if \\begin{equation} \\label{eq:8.1.16} \\lim_{n\\to\\infty}\\rho(u_n,u)=0. \\end{equation} In this case we say that $\\lim_{n\\to\\infty}u_n=u$. \\bbox"}
{"_id": "373", "title": "", "text": "A sequence $\\{u_n\\}$ in a metric space $(A,\\rho)$ is  a {\\it Cauchy sequence\\/}  if for every $\\epsilon>0$ there is an integer $N$ such that \\begin{equation} \\label{eq:8.1.17} \\rho(u_n,u_m)<\\epsilon\\mbox{\\quad and \\quad}m,n>N. \\end{equation}"}
{"_id": "374", "title": "", "text": "A metric space $(A,\\rho)$ is {\\it complete\\/}  if every Cauchy sequence in $A$ has a limit."}
{"_id": "375", "title": "", "text": "If $\\rho$ and $\\sigma$ are both metrics on a set $A$, then $\\rho$ and $\\sigma$ are {\\it equivalent \\/} \\hskip-.2em if there are positive constants  $\\alpha$  and $\\beta$ such that \\begin{equation} \\label{eq:8.1.18} \\alpha\\le\\frac{\\rho(x,y)}{\\sigma(x,y)}\\le\\beta \\mbox{\\quad for all \\quad}x,y\\in A\\mbox{\\quad such that \\quad}x\\ne y. \\end{equation}"}
{"_id": "376", "title": "", "text": "The {\\it diameter\\/} of a nonempty subset $S$ of $A$ is $$ d(S)=\\sup\\set{\\rho(u,v)}{u,\\, v\\in T}. $$ If $d(S)<\\infty$ then $S$ is {\\it bounded\\/}. \\bbox"}
{"_id": "377", "title": "", "text": "A set $T$  is {\\it compact\\/} if it has the Heine--Borel property."}
{"_id": "378", "title": "", "text": "A  set $T$  is {\\it totally bounded\\/}  if for every $\\epsilon>0$ there is a finite set $T_\\epsilon$ with the following property: if $t\\in T$, there is an $s\\in T_\\epsilon$ such that $\\rho(s,t)<\\epsilon$. We say that $T_\\epsilon$ is a {\\it finite $\\epsilon$-net for $T$\\/}. \\bbox"}
{"_id": "379", "title": "", "text": "A subset $T$ of $C[a,b]$ is {\\it uniformly bounded\\/} if there is a constant $M$ such that \\begin{equation} \\label{eq:8.2.6} |f(x)|\\le M \\mbox{\\quad if \\quad} a\\le x\\le b\\mbox{\\quad and \\quad} f\\in T. \\end{equation} A subset $T$ of $C[a,b]$ is {\\it equicontinuous\\/} if for each $\\epsilon>0$ there is a $\\delta>0$ such that \\begin{equation} \\label{eq:8.2.7} |f(x_1)-f(x_2)|\\le \\epsilon \\mbox{\\quad if \\quad} x_1,x_2\\in [a,b],\\quad |x_1-x_2|<\\delta,\\mbox{\\quad and \\quad}f\\in T. \\end{equation}"}
{"_id": "380", "title": "", "text": "We say that $$ \\lim_{u\\to \\widehat u}f(u)=\\widehat v $$ if $\\widehat u\\in\\overline D_f$ and for each $\\epsilon>0$ there is a $\\delta>0$  such that \\begin{equation} \\label{eq:8.3.1} \\sigma(f(u),\\widehat v)<\\epsilon\\mbox{\\quad if \\quad} u\\in D_f \\mbox{\\quad and \\quad} 0<\\rho(u,\\widehat u)<\\delta. \\end{equation}"}
{"_id": "381", "title": "", "text": "We say that $f$ is {\\it continuous\\/} at $\\widehat u$ if $\\widehat u\\in D_f$ and for each $\\epsilon>0$ there is a $\\delta>0$  such that \\begin{equation} \\label{eq:8.3.2} \\sigma(f(u),f(\\widehat u))<\\epsilon\\mbox{\\quad if \\quad} u\\in D_f\\cap N_\\delta(\\widehat u). \\end{equation} If $f$ is  continuous at every point of a set $S$, then $f$ is {\\it continuous on\\/} S. \\bbox"}
{"_id": "382", "title": "", "text": "A function $f$ is {\\it uniformly continuous\\/} on a subset $S$ of $D_f$ if for each $\\epsilon>0$ there is a $\\delta>0$ such that $$ \\sigma(f(u),f(v))<\\epsilon\\mbox{\\quad whenever \\quad} \\rho(u,v)<\\delta\\mbox{\\quad and \\quad}u,v\\in S. $$"}
{"_id": "383", "title": "", "text": "If $f:(A,\\rho)\\to (A,\\rho)$  is defined on all of $A$ and there is a constant $\\alpha$ in $(0,1)$ such that \\begin{equation} \\label{eq:8.3.7} \\rho(f(u),f(v))\\le\\alpha\\rho(u,v) \\mbox{\\quad for all\\quad} (u,v)\\in A\\times A, \\end{equation} then $f$ is  a {\\it contraction\\/} of $(A,\\rho)$. \\bbox"}