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\R$ stetig in $a \in D$ und $f(a) \neq 0$. Dann $\exists \delta > 0$ +mit $f(x) \neq 0$ für alle $x \in D \cap ]a-\delta, a + \delta[$, d.h. +es ex. Umgebungen von $a$, s.d. $f(x) \neq 0$ für alle Punkte +in dieser Umgebung. + +\begin{proof} + Wähle $\epsilon := \frac{|f(a)|}{4} > 0$. Dann $\exists \delta > 0$ + mit $|f(x) - f(a)| < \epsilon$ $\forall x \in D$ + mit $|x - a| < \delta$. \\ + \[ + \implies |f(x)| \ge |f(a)| - |\underbrace{f(x)-f(a)}_{< \epsilon}| > |f(a)| - \frac{|f(a)|}{4} = \frac{3}{4} |f(a)| > 0 + .\] $\forall x \in D$ mit $|x - a| < \delta$ +\end{proof} + +\begin{satz} + \begin{enumerate} + \item Es sei $f, g: D \to \R$ stetig in $a \in D$. Dann + sind $\lambda f + \mu g$ $\forall \lambda, \mu \in \R$, $f\cdot g$ und falls + $g(x) \neq 0$ $\forall x \in D$ $\frac{f}{g}$ stetig in $a$. + \item Sei $f$ stetig in $a \in D$ mit $f(D) \subset \overline{D} \subset \R$ + und $h\colon \overline{D} \to \R$ stetig in $f(a)$. Dann + ist die Komposition $(h\circ f)\colon D \to R, (h\circ f)(x) := h(f(x))$ + stetig in $a$. + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate} + \item folgt aus Rechenregeln für konvergente Folgen. z.B.: $(x_n)_{n \in \N}$ eine + Folge in $D$ mit $\lim_{n \to \infty} x_n = a$. Dann + \[ + (f+g)(a) = \lim_{n \to \infty} (f+g)(x_n) = \lim_{n \to \infty} f(x_n) + \lim_{n \to \infty} g(x_n) + = f(a) + g(a) + .\] + \item Sei $(x_n)_{n\in\N}$ Folge in $D$ mit $\lim_{n \to \infty} x_n = a$. Dann: + \[ + \lim_{n \to \infty} \underbrace{f(x_n)}_{y_n := f(x_n)} = b =: f(a) + .\] $(y_n)_{n\in\N}$ Folge in $\overline{D}$. Aus Stetigkeit von $h$ in $b$ folgt: + \[ + \lim_{n \to \infty} h(y_n) = h(b) \implies \lim_{n \to \infty} \underbrace{h(f(x_n))}_{(h \circ f)(x_n)} = h(f(a)) + .\] + \end{enumerate} +\end{proof} + +\begin{bsp} + \begin{enumerate} + \item Alle Polynome + \[ + f(x) := \sum_{k=0}^{n} a_k x^{k} + .\] sind stetig in $\R$. + \item Rationale Funktionen $\frac{f}{g}$ mit Polynomen + $f, g$ $(g \neq 0)$ sind stetig in $D := \{x \in \R \mid g(x) \neq 0\} $. + \item $f$ stetig in $D$, dann ist auch $|f|\colon D \to \R$ + stetig, als Komposition: $(|\cdot | \circ f)(x)$. + \item Heaviside Funktion ist stetig $\forall x \in R \setminus \{0\} $. + \item Dirichlet-Funktion $f\colon \R \to \R$ + \[ + f(x) = \begin{cases} + 1 & x \in \Q \\ + 0 & x \in \R \setminus \Q + \end{cases} + .\] ist in keinem Punkt stetig. + \begin{proof} + Sei $a \in \Q$. Es existiert eine Folge + \[ + x_n := a + \frac{\sqrt{2} }{n} \in \R \setminus \Q \text{ mit } \lim_{n \to \infty} x_n = a + .\] aber + \[ + \lim_{n \to \infty} \underbrace{f(x_n)}_{= 0} = 0 \neq 1 = f(a) + .\] Sei $a \in \R \setminus \Q$. Es ex. eine Folge von + rationalen Zahlen $x_n$ mit $\lim_{n \to \infty} x_n = a$. + $x_n \in \Q$ $\forall n \in \N$ (Konstruktion von reellen Zahlen) es gilt + \[ + \lim_{n \to \infty} \underbrace{f(x_n)}_{= 1} = 1 \neq 0 = f(a) + .\] + \end{proof} + \end{enumerate} +\end{bsp} + +\subsection{Weitere Eigenschaften stetiger Funktionen} + +\begin{definition}[offene, abgeschlossene, kompakte Mengen] + Eine Menge $D \subset \R$ heißt offen, falls $\forall x \in D$ + $\exists r > 0$ mit + \[ + B_r(x) := \; ]x - r, x + r[ \; \subset D + .\] d.h. jeder Punkt besitzt eine Umgebung, welche ganz + in $D$ liegt. + + Eine Menge $D \subset \R$ heißt abgeschlossen, falls + die Grenzwerte von konvergenten Folgen aus $D$ wieder in $D$ liegt, d.h. + $(x_n)_{n\in\N}$ in $D$ mit $\lim_{n \to \infty} x_n = x_0 \implies x_0 \in D$. + + $D \subset \R$ heißt kompakt, falls $D$ beschränkt und abgeschlossen + ist. (beschränkt $\stackrel{\text{def.}}{=}$ $\exists C > 0$ mit + $|x| < C$ $\forall x \in D$ ) +\end{definition} + +\begin{bsp} + \begin{enumerate} + \item $]0,1[$ ist offen. $\forall x \in \; ]0,1[$ setze + $r := \frac{1}{2} \text{min}\{x, 1 - x\} $. + Man kann zeigen, dass $B_r(x) \subset \; ]0, 1[$ + + $]0, 1[$ ist nicht abgeschlossen, weil $\left(\frac{1}{n}\right)_{n \in \N} \subset \; ]0, 1[$. + mit $\frac{1}{n} \to 0 \not\in \; ]0, 1[$. + \item $[0,1]$ ist kompakt. $x \in [0,1] \implies |x| \le 1 \implies [0,1]$ beschränkt. + + Sei $(x_n)_{n \in \N}$ in $[0,1]$ mit $\lim_{n \to \infty} x_n = x_0$. Dann + gilt + \[ + 0 \le x_n \le 1 \; \forall n \stackrel{\text{Sandwich}}{\implies} 0 \le x_0 \le 1 \implies x_0 \in [0,1] \quad \text{abgeschlossen} + .\] + + $[0,1]$ ist nicht offen, da $0 \in [0,1]$, aber $\underbrace{]-r, r[}_{B_r(0) \subset [0,1]} \quad \forall r > 0$ + \item $\R$ ist offen, abgeschlossen aber nicht kompakt. + \end{enumerate} +\end{bsp} + +\begin{lemma}[Folgenkompakt] + $D \subset \R$ ist kompakt genau dann wenn, alle Folgen $(x_n)_{n\in\N}$ in $D$ + eine konvergente Teilfolge enthalten mit Grenzwert in $D$. +\end{lemma} + +\begin{proof} + \begin{itemize} + \item ,,$\implies$ '' Sei $(x_n)_{n\in\N}$ eine Folge in $D$, + Folge $(x_n)_{n\in\N}$ beschränkt $\implies$ nach Satz + von Bolzano-Weierstraß existiert eine konvergente Teilfolge + $(x_{n_k})_{k\in\N} \subset D$ $\stackrel{D \text{ abgeschlossen}}{\implies} \lim_{k \to \infty} x_{n_k} \in D$ + \item ,,$\impliedby$'' Angenommen. $D$ ist unbeschränkt, d.h. + $\forall n \in \N$ $\exists x_n \in D$ mit $|x_n| \ge n$. + + Dann enthält $(x_n)_{n\in\N}$ keine konvergente Teilfolge, weil alle + Teilfolgen unbeschränkt sind. Widerspruch $\implies$ $D$ ist beschränkt. + + Bleibt zu zeigen: $D$ ist abgeschlossen. Sei $(x_n)_{n\in\N}$ in $D$ mit + $x_n \to x_0, n \to \infty$. Nach Voraussetzungen + existiert eine konvergente Teilfolge von $(x_n)_{n \in \N}$ mit Limes + in $D$. Da alle Teilfolgen ebenfalls gegen $x_0$ konvergieren folgt, dass + $x_0 \in D$. + \end{itemize} +\end{proof} + +\begin{satz}[Das stetige Bild kompakter Mengen ist kompakt] + Sei $f\colon D \to \R$ stetig mit $D \subset \R$ kompakt. Dann + ist $f(D) = \{ f(x) \mid x \in D\} $ kompakt. +\end{satz} + +\begin{proof} + Zu zeigen: $f(D)$ ist kompakt. Sei $(y_n)_{n\in\N}$ eine Folge + in $f(D)$. Dann ex. eine Folge $(x_n)_{n\in\N} \subset D$ mit + $f(x_n) = y_n$ $\forall n \in \N$ ($f$ stetig). + + $D$ kompakt $\implies$ $\exists $ Teilfolge $(x_{n_k})_{k \in\N}$ mit + $\lim_{k \to \infty} x_{n_k} = x_0 \in D $. + \[ + f \text{ stetig } \implies f(x_{n_k}) = \underbrace{y_{n_k}}_{\text{Teilfolge in }f(D)} \to f(x_0) \in f(D) + .\] +\end{proof} + +\begin{definition}[Supremum, Infimum, Maximum, Minimum reellwertiger Funktionen] + Sei $f \colon D \to \R$, $D \subset \R$. + + \begin{align*} + \operatorname{sup}_{x \in D} f(x) \quad &\text{kleinste obere Grenze der Bildmenge } B_f := \{f(x) \mid x \in D\} \\ + & := \text{sup } B_f := \text{min}\{\beta \in \R \mid y \le \beta \; \forall y \in B_f \} + .\end{align*} + \begin{align*} + \operatorname{inf}_{x \in D} f(x) \; := \text{inf } B_f := \text{min}\{\beta \in \R \mid y \le \beta \; \forall y \in B_f\} + .\end{align*} + Falls $B_f := f(D)$ beschränkt ist, dann $\exists $ inf und sup. + + $x_min \in D$ heißt Minimum, $x_max$ Maximum von $f$, falls + \[ + \begin{cases} + \text{inf } f(x) = f(x_{min}) =: \text{min } f(x) \\ + \text{sup } f(x) = f(x_{max}) =: \text{max } f(x) + \end{cases} + .\] +\end{definition} + +\begin{satz} + Stetige reelwertige Funktionen nehmen auf kompakten Mengen + ihr Minimum und Maximum an, d.h. + $f\colon D \to \R$ stetig, $D$ kompakt, dann ex. $x_{min}, x_{max} \in D$ mit \\ + $f(x_{min}) = \text{inf } \{f(x) \mid x \in D\} $ \\ + $f(x_{max}) = \text{sup } \{f(x) \mid x \in D\} $ +\end{satz} + +\begin{proof} + Folgt aus Satz. Zunächst $f(D)$ ist beschränkt, d.h. + dass Supremum und Infimum von $f(D)$ existieren. Nach Definition von + $s := \text{sup }\{f(x) \mid x \in D\} $ ex. eine Folge $(x_n)_{n\in\N}$ + in $D$ mit $f(x_n) \to s, n \to \infty$. $(x_n)_{n\in\N}$ hat + eine konvergente Teilfolge $(x_{n_k})_{k \in\N}$ mit + $x_{n_k} \to x_{max}$, $k \to \infty$, $x_{max} \in D$.\\ + $\implies$ $f(x_{n_k}) \to f(x_{max}), k \to \infty$\\ + $\implies$ Behauptung für Supremum +\end{proof} + +\begin{satz}[Zwischenwertsatz] + Sei $f\colon \underbrace{[a,b]}_{\text{kompakt}} \to \R$ stetig + mit $f(a) \neq f(b)$. + Dann gibt es zu jedem $y$ zwischen $f(a)$ und $f(b)$ ($*$) + mindestens ein $c \in [a,b]$ mit $f(c) = y$. + + $*$ d.h. $f(a) \le y \le f(b)$ falls $f(a) \le f(b)$, + sonst $f(b) \le y \le f(a)$) +\end{satz} + +\end{document}