From bdcb40e561a05ee171f78067980b0bd243dc8a49 Mon Sep 17 00:00:00 2001 From: Robert Date: Sat, 27 Mar 2021 20:04:45 +0100 Subject: [PATCH] finished continuity chapter --- .gitignore | 3 +- chapters/sections/continuity.tex | 411 +++++++++++++++++++++++++++ chapters/sections/seq_ser_limits.tex | 2 +- chapters/topo_of_metr_spaces.tex | 1 + script.loe | 0 script.pdf | Bin 712812 -> 749971 bytes script.tex | 2 + 7 files changed, 417 insertions(+), 2 deletions(-) create mode 100644 chapters/sections/continuity.tex delete mode 100644 script.loe diff --git a/.gitignore b/.gitignore index 4414c36..a59d1c7 100644 --- a/.gitignore +++ b/.gitignore @@ -4,4 +4,5 @@ *.log *.out *.gz -*.toc \ No newline at end of file +*.toc +*.loe \ No newline at end of file diff --git a/chapters/sections/continuity.tex b/chapters/sections/continuity.tex new file mode 100644 index 0000000..6791fe3 --- /dev/null +++ b/chapters/sections/continuity.tex @@ -0,0 +1,411 @@ +\documentclass[../../script.tex]{subfiles} +%! TEX root = ../../script.tex + +\begin{document} +\section{Continuity} +\begin{center} + \begin{tikzpicture} + \draw (-2, 0) ellipse (1cm and 2cm) node[above=1.4cm] (X) {$X$}; + \draw (2, 0) ellipse (1cm and 2cm) node[above=1.4cm] (Y) {$Y$}; + + \draw [->] (-0.9, 0.8) to [out=30, in=150] node[above] {$f$} (0.9, 0.8); + + \draw[fill] (-2, -0.2) circle [radius=1pt] node[above right] {$x$}; + \draw[fill] (2, -0.2) circle [radius=1pt] node[above right] {$y$}; + + \draw[dashed] (-2, -0.2) circle [radius=15pt] node[below=0.5cm] {$\oball[\delta](x)$}; + \draw[dashed] (2, -0.2) circle [radius=15pt] node[below=0.5cm] {$\oball(y)$}; + \end{tikzpicture} +\end{center} + +\begin{defi} + Let $(X, d_X), (Y, d_Y)$ be metric spaces. $f: x \rightarrow y$ is said to be continuous in $x \in X$ if + \[ + \forall \epsilon > 0 ~\exists \delta > 0: ~~\tilde{x} \in \oball[\delta](x) \implies f(\tilde{x}) \in \oball(f(x)) + \] + $f$ is said to be continuous is it is continuous in every point. +\end{defi} + +\begin{eg} + \begin{enumerate}[(i)] + \item Let $\metric$ be a metric space. + \begin{align*} + \idf: X &\longrightarrow X \\ + x &\longmapsto x + \end{align*} + is continuous (choose $\delta = \epsilon$). + + \item The function + \begin{align*} + f: \realn^2 &\longrightarrow \realn^2 \\ + (x, y) &\longmapsto (x, -y) + \end{align*} + is continuous. For $(\tilde{x}, \tilde{y}), (x, y) \in \realn^2$ we have + \begin{align*} + \norm{f(\tilde{x}, \tilde{y}) - f(x, y)}^2 &= \norm{(\tilde{x} - x, y - \tilde{y})}^2 = (\tilde{x} - x)^2 + (y - \tilde{y})^2 \\ + &= \norm{(\tilde{x}, \tilde{y}) - (x, y)}^2 + \end{align*} + + \item Consider + \begin{align*} + f: \realn^2 &\longrightarrow \realn \\ + (x, y) &\longmapsto \begin{cases} + 0, &x \cdot y = 0 \\ + 1, &x \cdot y \ne 0 + \end{cases} + \end{align*} + $f$ is non continuous in $(0, 0)$. + \end{enumerate} +\end{eg} + +\begin{rem} + \begin{enumerate}[(i)] + \item \[f \text{ continuous in } x \iff \forall \epsilon > 0 ~\exists \delta > 0: ~~f(\oball[\delta](x)) \subset \oball(f(x))\] + \item Continuity is a local property, this means if $x \in X$, $U$ a neighbourhood of $x$ and $f, g$ functions with $f \vert_U = g \vert_U$, then + \[ + f \text{ continuous} \iff g \text{ continuous} + \] + \end{enumerate} +\end{rem} + +\begin{thm} + Let $x_0 \in X$, $g: X \rightarrow Y$ and $f: Y \rightarrow Z$. If $g$ is continuous in $x_0$ and $f$ is continuous in $g(x_0)$, + then $f \circ g$ is continuous in $x_0$. + + \begin{center} + \begin{tikzpicture} + \draw (-4, 0) ellipse (1cm and 2cm) node[above=1.4cm] {$X$}; + \draw (0, 0) ellipse (1cm and 2cm) node[above=1.4cm] {$Y$}; + \draw (4, 0) ellipse (1cm and 2cm) node[above=1.4cm] {$Z$}; + + \draw [->] (-2.9, 0.8) to [out=30, in=150] node[above] {$g$} (-1.1, 0.8); + \draw [->] (1.1, 0.8) to [out=30, in=150] node[above] {$f$} (2.9, 0.8); + + \draw[fill] (-4, -0.2) ellipse [radius=1pt] node[below] {$x_0$}; + \draw[fill] (0, -0.2) ellipse [radius=1pt] node[below] {$g(x_0)$}; + \draw[fill] (4, -0.2) ellipse [radius=1pt] node[below] {$f(g(x_0))$}; + \end{tikzpicture} + \end{center} +\end{thm} +\begin{proof} + Since $f, g$ are continuous we know that + \begin{subequations} + \begin{align} + \forall \epsilon > 0 ~\exists \delta > 0&: ~~y \in \oball[\delta](g(x_0)) \implies f(y) \in \oball(f(g(x_0))) \\ + \forall \delta > 0 ~\exists \rho > 0&: ~~x \in \oball[\rho](x_0) \implies g(x) \in \oball[\delta](g(x_0)) + \end{align} + \end{subequations} + Then $\forall x \in \oball[\rho](x_0)$ we have + \begin{equation} + (f \circ g)(x_0) = f(g(x_0)) \in \oball(f(g(x_0))) + \end{equation} +\end{proof} + +\begin{defi}[Lipschitz continuity] + A function $f: X \rightarrow Y$ is said to be Lipschitz continuous if + \[ + \exists L > 0: ~~d_Y(f(x), f(y)) \le L \cdot D_X(x, y) + \] + $L$ is called Lipschitz constant. If $L = 1$, $f$ is called contraction. +\end{defi} + +\begin{eg} + Let $f, g: [0, 1] \rightarrow \realn$. + \begin{align*} + f(x) = x^2 && g(x) = \sqrt{x} + \end{align*} + $f$ is Lipschitz continuous, $g$ is not. +\end{eg} + +\begin{thm} + Every Lipschitz continuous function is continuous. +\end{thm} +\begin{proof} + Let $f: X \rightarrow Y$ be Lipschitz continuous, with Lipschitz constant $L$. Let $\epsilon > 0$, + then for $x \in \oball[\frac{\epsilon}{L}](x_0)$ + \begin{equation} + d(f(x), f(x_0)) \le L \cdot d(x, x_0) < \epsilon + \end{equation} + Thus, $f$ is continuous in $x_0$, and since we chose an arbitrary $x_0$, $f$ is continuous everywhere. +\end{proof} + +\begin{eg} + \begin{enumerate}[(i)] + \item Consider + \begin{align*} + \pi_i: \field^n &\longrightarrow \field \\ + (x_1, x_2, \cdots, x_n) &\longmapsto x_i + \end{align*} + Then + \[ + \abs{\pi_i(x) - \pi_i(y)} = \abs{x_i - y_i} \le \norm{x - y} + \] + So $\pi_i$ is a contraction. + + \item Let $(X, d), (X \times X, d_{X \times X})$ be metric spaces. Then + \begin{align*} + d: X \times X &\longrightarrow \realn \\ + (x, y) &\longmapsto d(x, y) + \end{align*} + is a contraction. Let $x_1, x_2, y_1, y_2 \in X$ and apply the triangle inequality + \[ + d(x_1, y_1) \le d(x_1, x_2) + d(x_2, y_1) \le d(x_1, x_2) + d(y_2, y_1) + d(x_2, y_2) + \] + This implies + \begin{align*} + \abs{d(x_1, y_1) - d(x_2, y_2)} &\le d(x_1, x_2) + d(y_1, y_2) \\ + &= d_{X \times X} ((x_1, x_2), (y_1, y_2)) + \end{align*} + which means the metric is continuous. + + \item Analogously, this works for $\dnorm$. + \end{enumerate} +\end{eg} + +\begin{thm} + Let $f: X \rightarrow Y$. + \[ + f \text{ is continuous in } x \in X \iff \substack{x \text{ is an isolated point in } X \\ \text{or } \limes{\tilde{x}}{x} f(\tilde{x}) = f(x)} + \] +\end{thm} +\begin{proof} + Let $f$ be continuous in $x \in X$. If $x$ is an isolated point there is nothing to show, so let $x$ be a limit point. Then + \begin{equation} + \forall \epsilon > 0 ~\exists \delta > 0: ~~f(\tilde{x}) \in \oball(f(x)) ~~\forall \tilde{x} \in \oball[\delta](x) + \end{equation} + Now let $x$ be an isolated point, i.e. $\exists \delta > 0$ such that $\oball[\delta](x) = \set{x}$. Then + \begin{equation} + f(\oball[delta](x)) = \set{f(x)} \subset \oball(f(x)) ~~\forall \epsilon > 0 + \end{equation} + If $x$ is a limit point and $\limes{\tilde{x}}{x} f(\tilde{x}) = f(x)$, then let $\epsilon > 0$ + \begin{equation} + \exists \delta > 0: ~~f(\dot{\oball[\delta]}(x)) \subset \oball(f(x)) + \end{equation} + This then implies + \begin{equation} + f(\oball[\delta]) \subset \oball(f(x)) + \end{equation} +\end{proof} + +\begin{cor} + \[ + f: X \rightarrow Y \text{ continuous in } x \in X \iff \forall \anyseqdef{X}: ~~f(x_n) \conv{x_n \rightarrow x} f(x) + \] + This means, for continuous $f$ we have + \[ + \limn f(x_n) = f(\limn x_n) + \] +\end{cor} + +\begin{cor} + Let $f_1, \cdots, f_n: \realn^m \rightarrow \real$. Then define + \begin{align*} + f: \realn^m &\longrightarrow \realn^n \\ + x &\longmapsto \left( f_1(x), f_2(x), \cdots, f_n(x) \right) + \end{align*} + $f$ is continuous if and only if $f_1, \cdots, f_n$ are continuous. +\end{cor} + +\begin{cor} + Let $f, g: X \rightarrow \realn$ be continuous in $x \in X$. Then + \begin{align*} + f + g && f \cdot g + \end{align*} + are continuous in $x$, and if $g(x) \ne 0$ then + \[ + \frac{f}{g} + \] + is also continuous in $x$. +\end{cor} + +\begin{eg} + Let $\eta = (\eta_1, \cdots, \eta_n) \in \natn_0^n$ and $x \in \field^n$. Define + \[ + x^{\eta} = x_1^{\eta_1} \cdot x_2^{\eta_2} \cdot x_3^{\eta_3} \cdot \cdots \cdot x_n^{\eta_n} + \] + $\eta$ is called multi index. We set + \[ + \abs{\eta} := \eta_1 + \eta_2 + \eta_3 + \cdots + \eta_n + \] + Let $c_{\eta} \in \field ~~\forall \eta$ with $\abs{\eta} \le N ~~N \in \natn$. Then we call + \begin{align*} + f: \field^n &\longrightarrow \field \\ + x &\longmapsto \sum_{|\eta| \le N} c_{\eta} \cdot x^{\eta} + \end{align*} + a polynomial with $n$ variables. Such polynomials are continuous. Example: + \[ + (x_1, x_2) \longmapsto x_1^2 + x_2^2 + x_1^9 + x_2^{17} + \] +\end{eg} + +\begin{rem} + In the context of polynomials (and power series) we define + \[ + 0^0 = 1 + \] + Reminder: If $f: X \rightarrow Y$ and $U \subset Y$ then $\inv{f}(U)$ is said to be the preimage of $U$ under $f$. + It's the set of all points of $X$ that get mapped to $U$. + \[ + \inv{f}(U) = \set[f(x) \in U]{x \in X} + \] +\end{rem} + +\begin{thm} + Let $f: X \rightarrow Y$ + \begin{enumerate}[(i)] + \item + \[ + f \text{ is continuous in } x \iff \substack{\inv{f}(U) \text{ is a neighbourhood of} \\ x ~~\forall U \text{neighbourhood of } f(x)} + \] + + \item + \[ + f \text{ is continuous} \iff \inv{f}(O) \text{ is open } \forall O \subset Y \text{ open} + \] + + \item + \[ + f \text{ is continuous} \iff \inv{f}(C) \text{ is closed } \forall C \subset Y \text{ closed} + \] + \end{enumerate} +\end{thm} +\begin{proof} + We will prove (i). Let $U$ be a neighbourhood of $f(x)$, i.e. + \begin{equation} + \exists \epsilon > 0: ~~\oball(f(x)) \subset U + \end{equation} + Since $f$ is continuous + \begin{equation} + \exists \delta > 0: ~~f(\oball[\delta](x)) \subset \oball(f(x)) + \end{equation} + which in turn means + \begin{equation} + \oball[\delta](x) \subset \inv{f}(\oball(f(x))) \subset \inv{f}(U) + \end{equation} + so $\inv{f}(U)$ is a neighbourhood of $f(x)$. Now let $\epsilon > 0$. + Since $\oball(f(x))$ is a neighbourhood of $f(x)$, $\inv{f}(\oball(f(x)))$ is a neighbourhood of $x$. This means + \begin{equation} + \exists \delta > 0: ~~\oball[\delta](x) \subset \inv{f}(\oball(f(x))) + \end{equation} + Thus $f(\oball[\delta](x)) \subset \oball(f(x))$ which means $f$ is continuous in $x$. + + (ii) and (iii) are left to the reader. +\end{proof} + +\begin{defi}[Subsequences and (sequential) compactness] + Let $\metric$ be a metric space, and $\anyseqdef{X}$, $(n_k) \subset \natn$ are strictly monotonically increasing. + Then $(x_{n_k})$ is said to be a subsequence of $(x_n)$. + + A subset $A \subset X$ is said to be (sequentially) compact, if every sequence $\seq{x} \subset A$ has a subsequence convergent in $A$. +\end{defi} + +\begin{rem} + If $\seq{x}$ converges to $x \in X$, then every subsequence of $\seq{x}$ converges to $x$. However, consider + \[ + \seq{x} = (-1)^n + \] + This sequence doesn't converge, but the subsequences $(x_{2n})$ and $(x_{2n + 1})$ converge to (different) values. +\end{rem} + +\begin{eg} + Let $X = \realn$, then $(0, 1)$ and $\natn$ are not compact. Because + \begin{align*} + (x_n = \frac{1}{n}) \subset (0, 1) && (x_n = n) \subset \natn + \end{align*} + have no convering subsequences. +\end{eg} + +\begin{thm} + \[ + A \subset \realn^n \text{ is compact} \iff A \text{ closed and bounded} + \] +\end{thm} +\begin{proof} + Assume $A$ is not closed, i.e. for $x \in \boundary{A} \setminus A$ + \begin{equation} + \exists \anyseqdef{A} \text{ with } x_n \conv{} x + \end{equation} + Every subequence of $\seq{x}$ converges to $x$, but $x \ne A$. From this follows that $A$ is not compact. + Assume $A$ is not bounded, i.e. $A \setminus \oball[n](0) \ne \varnothing ~~\forall n \in \natn$. Now choose + $\anyseqdef{A}$ such that $\norm{\seq{x}} \ge n$. $\seq{x}$ cannot have a convergent subsequence, because on the one hand + for $(x_{n_k})$ convergent to $x$ we have $\norm{x_{n_k}} \rightarrow \norm{x}$, but on the other hand + $\norm{x_{n_k}} \ge n_k \longrightarrow \infty$. This proves the "$\implies$" direction, to prove the inverse, consider the case $n = 1$: + Let $A \subset \realn$ be bounded and closed. Then + \begin{equation} + \exists K > 0: ~~A \subset I_1 = [-K, K] + \end{equation} + Let $\anyseqdef{A}$ be a sequence. We recursively define more intervals. Let $I_k = [a, b)$ such that $x_n \in I_k$ for infinitely many $n \in \natn$. + Half the interval: + \begin{subequations} + \begin{align} + I_{k+1} = \left[a, \frac{b - a}{2}\right) && \text{or} && I_{k+1} = \left[\frac{b - a}{2}, b\right) + \end{align} + \end{subequations} + such that $x_n \in I_{k+1}$ for infinitely many $n \in \natn$. By doing this we are creating a sequence of nested intervals of length $K \cdot 2^{-k + 2}$. + Now set $n_1 = 1$, and then recursively define + \begin{equation} + n_{k+1} > \max\set{n_1, \cdots, n_k} \text{ and } x_{n_{k+1}} \in I_{k + 1} + \end{equation} + We now need to show that $(x_{n_k})$ is convergent. Apply the Cauchy criterion: For $l > k$ we know that $x_{n_k}$ and $x_{n_l} \in I_k$, i.e. + \begin{equation} + \abs{x_{n_k} - x_{n_l}} \le K \cdot 2^{-k + 2} \conv{k \rightarrow \infty} 0 + \end{equation} + This means, $x_{n_k}$ is a Cauchy sequence, so it converges to $x \in \realn$. Since $A$ is closed, we have $x \in A$. +\end{proof} + +\begin{thm} + Continuous mappings map compact sets to compact sets. +\end{thm} +\begin{proof} + Let $f: X \rightarrow Y$ be continuous and $A \subset X$ compact. Let $\seq{x} \subset f(A)$. We need to show that $\seq{x}$ has a convergent subsequence. + We know that + \begin{equation} + \exists \anyseqdef[y]{A}: ~~x_n = f(y_n) + \end{equation} + Since $A$ is compact, there must be subsequences $(y_{n_k})$ with $y_{n_k} \conv{k \rightarrow \infty} y \in A$. Because of the continuity of $f$, we have + \begin{equation} + \underbrace{f(y_{n_k})}_{x_{n_k}} \conv{} f(y) \in f(A) + \end{equation} + Thus, $f(A)$ is compact. +\end{proof} + +\begin{rem} + Let $f: \realn^n \rightarrow \realn^n$ be a continuous mapping. $f$ maps closed, bounded sets to closed, bounded sets. + In general, closed sets are NOT mapped to closed sets, and bounded sets are NOT mapped to bounded sets. + + Example: $f: (0, \infty) \rightarrow \realn, ~~x \mapsto \inv{x}$ + \begin{align*} + f(\underbrace{(0, 1)}_{\text{bounded}}) = \underbrace{(1, \infty)}_{\text{unbounded}} + && f(\underbrace{[1, \infty]}_{\text{closed}}) = \underbrace{(0, 1]}_{\text{not closed}} + \end{align*} +\end{rem} + +\begin{cor} + Let $A \subset \realn^n$ be compact and $f: A \rightarrow \realn$ continuous. Then $f$ assumes its maximum on $A$. I.e. + \[ + \exists x \in A: ~~f(y) \le f(x) ~~\forall y \in A + \] +\end{cor} +\begin{proof} + $f(A)$ is compact, so it's closed and bounded. We want to show that compact subsets $K$ of $\realn$ have a maximum $M := \sup K$ + such that $x_n \conv{} M$. Since $K$ is closed we know that $M \in \field$, so $M$ is a maximum. + Especially, $\exists z \in f(A)$ maximum and $\exists x \in A$ with $f(x) = z$ +\end{proof} + +\begin{thm} + Let $A \subset \realn^n, B \subset \realn^m$ be compact subsets and $f: A \rightarrow B$ a bijective, continuous mapping. + Then $\inv{f}$ is also continuous. +\end{thm} +\begin{proof} + Define $g := \inv{f}$. $g$ is also bijective and maps $B \rightarrow A$. Let $C \subset A$ be closed. + Since $A$ is bounded, $C$ is also bounded. Thus, $f(C)$ is also compact (i.e. bounded and closed), and we have + \begin{equation} + \begin{split} + f(C) &= \set[x \in C]{f(x) \in B} \\ + &= \set[g(y) \in C]{f(g(y)) \in B} \\ + &= \set[g(y) \in C]{y \in B} = \inv{g}(C) + \end{split} + \end{equation} + So $\inv{g}(C)$ is bounded, and since $C$ was an arbitrary closed set, $g$ is also continuous. +\end{proof} +\end{document} \ No newline at end of file diff --git a/chapters/sections/seq_ser_limits.tex b/chapters/sections/seq_ser_limits.tex index 362870a..a6338c3 100644 --- a/chapters/sections/seq_ser_limits.tex +++ b/chapters/sections/seq_ser_limits.tex @@ -1,4 +1,4 @@ -\documentclass[../../script.tex] {subfiles} +\documentclass[../../script.tex]{subfiles} %! 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