Theorems

Mean value theorem and Cauchy's mean value theorem (Tag T-OXNTU)

Let \(a<b\) and let \(f\colon [a, b] \to \RR\) be continuous on \([a,b]\) and differentiable on \((a, b)\).

1. There exists \(\xi \in (a, b)\) such that \[ f(b) - f(a) = f'(\xi)(b-a). \]

2. If \(g\colon [a,b]\to \RR\) is also continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists \(\xi\in(a,b)\) such that \[ \qty{ f(b) - f(a) } g'(\xi) = \qty{g(b) - g(a)} f'(\xi). \] [@Rud76].

Geometrically, (2) says that the plane curve \(t\mapsto(f(t),g(t))\), \(t\in[a,b]\), has a tangent vector \((f'(\xi),g'(\xi))\) at some interior parameter \(\xi\) that is parallel to the chord from \((f(a),g(a))\) to \((f(b),g(b))\).

Taylor remainders in Lagrange, Cauchy, and integral form (Tag T-2R7PC)

Let \(n\geq1\), \(p\in\RR\), \(\delta>0\), and let \(f\colon(p-\delta,p+\delta)\to\RR\) be \(n\) times differentiable. For \(x\in(p-\delta,p+\delta)\) put \[ R_n(x)\coloneqq f(x)-\sum_{k=0}^{n-1} \frac{f^{(k)}(p)}{k!}(x-p)^k . \] Let \(x\in(p-\delta,p+\delta)\) with \(x\neq p\).

1. (Lagrange form) There exists \(\xi\) strictly between \(p\) and \(x\) such that \[ R_n(x)=\frac{f^{(n)}(\xi)}{n!}(x-p)^n . \]

2. (Cauchy form) There exists \(\eta\) strictly between \(p\) and \(x\) such that \[ R_n(x)=\frac{f^{(n)}(\eta)}{(n-1)!}(x-\eta)^{n-1}(x-p) . \]

3. (Integral form) If moreover \(f^{(n)}\) is continuous on \((p-\delta,p+\delta)\), then \[ R_n(x)=\int_p^x \frac{f^{(n)}(t)}{(n-1)!}(x-t)^{n-1}\,dt . \]

Part 1 is in [@Rud76].

Characterizations of compact subsets of a metric space (Tag T-YOZX6)

Let \((X,d)\) be a metric space and let \(E\subseteq X\), with the metric induced by \(d\). The following are equivalent [@Fol13]:

1. \(E\) is complete and totally bounded: for every \(\varepsilon>0\), \(E\) is covered by finitely many balls of radius \(\varepsilon\).

2. \(E\) is sequentially compact: every sequence in \(E\) has a subsequence that converges to a point of \(E\).

3. \(E\) is compact: every cover of \(E\) by open subsets of \(X\) has a finite subcover.

Heine--Cantor theorem (Tag T-O4UD3)

Let \(X\) and \(Y\) be metric spaces with \(X\) compact. Every continuous function \(f\colon X\to Y\) is uniformly continuous [@Mun00]; [@Rud76].

Heine--Borel theorem (Tag T-QPTHZ)

A subset \(X\subseteq \RR^n\) is compact if and only if \(X\) is closed and bounded.

$\RR$ is not a countable union of nowhere dense sets (Tag T-7FJFK)

\(\RR\) is a Baire space. In particular, \(\RR\) is not a countable union of nowhere dense subsets of \(\RR\).

Egorov's theorem (Tag FT-OGS76)

Let \(m\) be Lebesgue measure on \(\RR^n\), let \(E\subseteq \RR^n\) be Lebesgue measurable with \(m(E) < \infty\), and let \(f_n\colon E\to\RR\) for \(n\geq1\) and \(f\colon E\to\RR\) be measurable with \(f_n \to f\) almost everywhere on \(E\). Then for every \(\varepsilon > 0\) there exists a closed set \(F\subseteq E\) such that \(m(E\setminus F)<\varepsilon\) and \(f_n\to f\) uniformly on \(F\).

Lusin's theorem (Tag T-CGFCU)

Let \(E\subseteq\RR^n\) be Lebesgue measurable with \(m(E)<\infty\), and let \(f\colon E\to\RR\) be measurable. For every \(\varepsilon>0\) there exists a closed set \(F_\varepsilon\subseteq\RR^n\) with \[ F_\varepsilon \subseteq E \quad\text{and}\quad m(E \setminus F_\varepsilon) \leq \varepsilon \] such that the restriction \(\ro{f}{F_\varepsilon}\colon F_\varepsilon\to\RR\) is continuous.

Equivalent characterizations of continuity (Tag T-RA-WORKSHOP-D4-3-1)

Let \(E\subseteq\RR^n\) and let \(f\colon E\to\RR^m\). The following are equivalent:

1. \(f\) is continuous on \(E\), with the subspace topology on \(E\).

2. For every \(x\in E\) and every \(\varepsilon>0\) there exists \(\delta>0\) such that \(\norm{f(y)-f(x)}<\varepsilon\) for all \(y\in E\) with \(\norm{x-y}<\delta\).

3. For every sequence \((x_n)_{n\geq1}\) in \(E\) converging to a point \(x\in E\), \(\lim_{n\to\infty}f(x_n)=f(x)\).

4. For every open set \(G\subseteq\RR^m\), the set \(\inverseof{f}(G)\) is open in \(E\) with the subspace topology. [@Rud76].

Continuous images of compact sets and the extreme value theorem (Tag T-RA-WORKSHOP-D4-3-2)

Let \(K\subseteq\RR^n\) be compact and let \(f\colon K\to\RR^m\) be continuous. Then \(f(K)\) is compact. In particular, if \(a<b\) and \(f\colon[a,b]\to\RR\) is continuous, then there exist \(p,q\in[a,b]\) such that \[ f(p)=\sup_{x\in[a,b]}f(x)\qquad\text{and}\qquad f(q)=\inf_{x\in[a,b]}f(x). \] [@Rud76].

Continuous functions on compact sets are uniformly continuous (Tag T-RA-WORKSHOP-D4-3-3)

Let \(K\subseteq\RR^n\) be compact and let \(f\colon K\to\RR^m\) be continuous. Then \(f\) is uniformly continuous on \(K\). [@Rud76].

Intermediate value theorem (Tag T-RA-WORKSHOP-D4-3-4)

Let \(a<b\) and let \(f\colon[a,b]\to\RR\) be continuous with \(f(a)<f(b)\). For every \(y\in\RR\) with \(f(a)<y<f(b)\) there exists \(c\in(a,b)\) such that \(f(c)=y\). [@Rud76].

Discontinuities of monotone functions (Tag T-RA-WORKSHOP-D4-3-5)

Let \(a<b\) and let \(f\colon(a,b)\to\RR\) be monotone. Then the set \(E\subseteq(a,b)\) of points at which \(f\) is discontinuous is at most countable. Moreover, every \(x\in E\) is a jump discontinuity: the one-sided limits \[ f(x-)\coloneqq\lim_{t\to x^-}f(t)\qquad\text{and}\qquad f(x+)\coloneqq\lim_{t\to x^+}f(t) \] both exist. [@Rud76].

Fermat's stationary-point theorem (Tag T-RA-WORKSHOP-D5-4-1)

Let \(a<b\), let \(f\colon[a,b]\to\RR\), and let \(c\in(a,b)\). If \(f\) has a local maximum or a local minimum at \(c\) and \(f\) is differentiable at \(c\), then \(f'(c)=0\). [@Rud76].

Mean value theorem (Tag T-RA-WORKSHOP-D5-4-2)

Let \(a<b\) and let \(f\colon[a,b]\to\RR\) be continuous on \([a,b]\) and differentiable on \((a,b)\). Then there exists \(\xi\in(a,b)\) such that \[ \frac{f(b)-f(a)}{b-a}=f'(\xi). \] [@Rud76].

Taylor's theorem with Lagrange remainder (Tag T-RA-WORKSHOP-D5-4-3)

Let \(a<b\), \(n\geq0\), and \(c\in[a,b]\). Suppose \(f\colon[a,b]\to\RR\) is \(n\) times continuously differentiable on \([a,b]\) and \(f^{(n+1)}\) exists on \((a,b)\). Let \[ P_n(x)\coloneqq\sum_{k=0}^{n}\frac{f^{(k)}(c)}{k!}(x-c)^k =f(c)+f'(c)(x-c)+\cdots+\frac{f^{(n)}(c)}{n!}(x-c)^n . \] Then for every \(x\in[a,b]\) with \(x\neq c\) there exists \(\xi\) strictly between \(c\) and \(x\) such that \[ f(x)=P_n(x)+\frac{f^{(n+1)}(\xi)}{(n+1)!}(x-c)^{n+1}. \] In particular, if \(M\geq0\) and \(\abs{f^{(n+1)}(t)}\leq M\) for all \(t\in(a,b)\), then \[ \abs{f(x)-P_n(x)}\leq\frac{M\abs{x-c}^{n+1}}{(n+1)!}\quad\text{for all } x\in[a,b]. \] [@Rud76].

Riemann's condition for Riemann--Stieltjes integrability (Tag T-RA-WORKSHOP-D6-5-1)

Let \(a<b\), let \(\alpha\colon[a,b]\to\RR\) be monotonically increasing, and let \(f\colon[a,b]\to\RR\) be bounded. For a partition \(P=\theset{a=x_0<x_1<\cdots<x_k=b}\) of \([a,b]\), put \(\Delta\alpha_i\coloneqq\alpha(x_i)-\alpha(x_{i-1})\) and \[ U(P,f,\alpha)\coloneqq\sum_{i=1}^k\qty{\sup_{[x_{i-1},x_i]}f}\Delta\alpha_i,\qquad L(P,f,\alpha)\coloneqq\sum_{i=1}^k\qty{\inf_{[x_{i-1},x_i]}f}\Delta\alpha_i . \] Then \(f\in\mathcal R(\alpha)\), that is, \(f\) is Riemann–Stieltjes integrable with respect to \(\alpha\) on \([a,b]\), if and only if for every \(\varepsilon>0\) there exists a partition \(P\) of \([a,b]\) such that \[ U(P,f,\alpha)-L(P,f,\alpha)<\varepsilon. \] [@Rud76].

Riemann--Stieltjes integration by parts (Tag T-RA-WORKSHOP-D6-5-2)

Let \(a<b\) and let \(f,\alpha\colon[a,b]\to\RR\). Write \(f\in\mathcal R(\alpha)\) if there is \(A\in\RR\) such that for every \(\varepsilon>0\) there is a partition \(P_\varepsilon\) of \([a,b]\) with \[ \abs{\sum_{i=1}^k f(t_i)\qty{\alpha(x_i)-\alpha(x_{i-1})}-A}<\varepsilon \] for every partition \(P=\theset{a=x_0<\cdots<x_k=b}\) refining \(P_\varepsilon\) and every choice of \(t_i\in[x_{i-1},x_i]\); then \(\int_a^b f\,d\alpha\coloneqq A\). If \(f\in\mathcal R(\alpha)\) on \([a,b]\), then \(\alpha\in\mathcal R(f)\) on \([a,b]\) and \[ \int_a^b f\,d\alpha=f(b)\alpha(b)-f(a)\alpha(a)-\int_a^b\alpha\,df. \] [@Apo74].

Reduction of a Stieltjes integral to a Riemann integral (Tag T-RA-WORKSHOP-D6-5-3)

Let \(a<b\), let \(\alpha\colon[a,b]\to\RR\) be monotonically increasing and differentiable with \(\alpha'\) Riemann integrable on \([a,b]\), and let \(f\colon[a,b]\to\RR\) be bounded. Then \(f\in\mathcal R(\alpha)\) on \([a,b]\) if and only if \(f\alpha'\) is Riemann integrable on \([a,b]\), and in that case \[ \int_a^b f\,d\alpha=\int_a^b f(x)\alpha'(x)\,dx. \] [@Rud76].

First fundamental theorem of calculus (Tag T-RA-WORKSHOP-D6-5-4)

Let \(a<b\), let \(f\colon[a,b]\to\RR\) be Riemann integrable on \([a,b]\), and define \(F\colon[a,b]\to\RR\) by \[ F(x)\coloneqq\int_a^x f(t)\,dt. \] Then \(F\) is continuous on \([a,b]\). If \(f\) is continuous at \(x_0\in[a,b]\), then \(F\) is differentiable at \(x_0\), one-sidedly if \(x_0\in\theset{a,b}\), and \(F'(x_0)=f(x_0)\). [@Rud76].

Mean value theorem for integrals (Tag T-RA-WORKSHOP-D6-5-5)

Let \(a<b\) and let \(f\colon[a,b]\to\RR\) be continuous. Then there exists \(c\in(a,b)\) such that \[ \frac{1}{b-a}\int_a^b f(x)\,dx=f(c). \] [@Apo74].

Uniform limits of continuous functions are continuous (Tag T-RA-WORKSHOP-D7-6-1)

Let \(E\) be a metric space, and let \(f_n\colon E\to\RR\) for \(n\geq1\) and \(f\colon E\to\RR\) be functions. If each \(f_n\) is continuous on \(E\) and \(f_n\to f\) uniformly on \(E\), then \(f\) is continuous on \(E\). [@Rud76].

Weierstrass M-test (Tag T-RA-WORKSHOP-D7-6-2)

Let \(E\) be a set, let \(f_n\colon E\to\RR\) for \(n\geq1\), and let \((M_n)_{n\geq1}\) be a sequence of real numbers such that \(\abs{f_n(x)}\leq M_n\) for all \(n\geq1\) and all \(x\in E\). If \(\sum_{n=1}^{\infty}M_n\) converges, then \(\sum_{n=1}^{\infty}f_n\) converges uniformly on \(E\). [@Rud76].

Uniform limits of Riemann--Stieltjes integrable functions (Tag T-RA-WORKSHOP-D7-6-3)

Let \(a<b\), let \(\alpha\colon[a,b]\to\RR\) be monotonically increasing, and let \(\mathcal R(\alpha)\) denote the set of bounded functions \([a,b]\to\RR\) that are Riemann–Stieltjes integrable with respect to \(\alpha\) on \([a,b]\). Let \(f_n\in\mathcal R(\alpha)\) for \(n\geq1\), and let \(f\colon[a,b]\to\RR\). If \(f_n\to f\) uniformly on \([a,b]\), then \(f\in\mathcal R(\alpha)\), the limit \(\lim_{n\to\infty}\int_a^b f_n\,d\alpha\) exists, and \[ \lim_{n\to\infty}\int_a^b f_n\,d\alpha=\int_a^b f\,d\alpha. \] [@Rud76].

Uniform convergence of derivatives and differentiability of the limit (Tag T-RA-WORKSHOP-D7-6-4)

Let \(a<b\), and let \(f_n\colon[a,b]\to\RR\) for \(n\geq1\) be differentiable on \([a,b]\). Suppose that \(\{f_n(x_0)\}\) converges for some \(x_0\in[a,b]\) and that \(\{f_n'\}\) converges uniformly on \([a,b]\). Then \(\{f_n\}\) converges uniformly on \([a,b]\) to a function \(f\colon[a,b]\to\RR\), \(f\) is differentiable on \([a,b]\), and \[ f'(x)=\lim_{n\to\infty}f_n'(x)\qquad\text{for all }x\in[a,b]. \] [@Rud76].

Arzelà--Ascoli theorem for $C(K,\RR^m)$ (Tag T-RA-WORKSHOP-D7-6-5)

Let \(n,m\geq1\), let \(K\subseteq\RR^n\) be compact, and equip \(C(K,\RR^m)\) with the metric \(d(f,g)\coloneqq\sup_{x\in K}\norm{f(x)-g(x)}\). A subset \(\mathcal F\subseteq C(K,\RR^m)\) is compact if and only if \(\mathcal F\) is closed, bounded, and equicontinuous at every point of \(K\), that is, \[ \forall x\in K\quad\forall\varepsilon>0\quad\exists\delta>0\quad\forall y\in K\quad\forall f\in\mathcal F:\quad\norm{x-y}<\delta\implies\norm{f(x)-f(y)}<\varepsilon. \] [@Mun00].

Weierstrass approximation and Stone--Weierstrass theorems (Tag T-RA-WORKSHOP-D7-6-7)

1. Let \(a<b\) and let \(f\colon[a,b]\to\CC\) be continuous. Then there exist polynomials \(P_n\) with complex coefficients such that \(P_n\to f\) uniformly on \([a,b]\). If \(f\) is real-valued, the \(P_n\) may be taken with real coefficients.

2. Let \(K\) be a compact metric space, let \(C(K)\) be the algebra of continuous functions \(K\to\CC\), and let \(\mathcal A\subseteq C(K)\) be a subalgebra, that is, a complex vector subspace closed under pointwise multiplication. Suppose that

• \(\mathcal A\) is self-adjoint: \(\bar g\in\mathcal A\) for every \(g\in\mathcal A\);

• \(\mathcal A\) separates points of \(K\): for all \(x\neq y\) in \(K\) there exists \(g\in\mathcal A\) with \(g(x)\neq g(y)\);

• \(\mathcal A\) vanishes at no point of \(K\): for every \(x\in K\) there exists \(g\in\mathcal A\) with \(g(x)\neq0\).

Then for every \(f\in C(K)\) there exist \(f_n\in\mathcal A\) such that \(f_n\to f\) uniformly on \(K\). [@Rud76].