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z(;g{!l0ybEPQCcKm$%)Q7f`f}y|KeQ&<6Pshs$csPd1#=jIcW4w8JtJ`1jIHhi-lg zwEgZkON(ZXAo7(v=@pkW&fbbm`KB*96-bsyn<_5ylFR~O4b4)CQli; 0 + .\] +\end{definition} + +\begin{bsp} + \begin{enumerate} + \item $a \in D \implies$ $a$ Berührpunkt von $D$ + \item $D = ]0,1[$, $0$ ist Berührpunkt von $D$, denn + $\forall \delta > 0$ $]-\delta, \delta[ \; \cap \; ]0,1[ \; \neq \; \emptyset$, + da $\delta > 0$ + \item $D = [1,2], 0$ ist kein Berührpunkt von $D$, + denn z.B. für $\delta = \frac{1}{2}$ : + \[ + ]-\frac{1}{2}, \frac{1}{2}[ \cap [1,2] = \emptyset + .\] + \end{enumerate} +\end{bsp} + +\begin{lemma}[Äquivalente Definition von Berührpunkten] + $a$ ist ein Berührpunkt von $D$ $\iff$ $\exists $ Folge $(a_n)_{n\in\N} \subset D$ mit + $\lim_{n \to \infty} a_n = a$ +\end{lemma} + +\begin{proof} + durch Behauptung +\end{proof} + +\begin{definition}[Grenzwert bei Funktionen] + \begin{enumerate} + \item Sei $f: D \to \R$, $D \subset \R$ und $x_0$ sei ein + Berührpunkt von $D$. $f$ hat in $x_0$ den + Grenzwert (oder limes), $y_0 \in \R$, falls + \[ + \forall \epsilon > 0 \quad \exists \delta > 0\colon |f(x) - y_0| < \epsilon \quad \forall x \in D, |x - x_0| < \delta + .\] + Schreibweise: + \[ + \lim_{x \to x_0} f(x) = y_0 + .\] $y_0$ ist eindeutig bestimmt. + + $y_0$ kann von $D$ abhängig sein und man schreibt daher + zur Verdeutlichung ein $x \in D$ darunter. + \item Sei $x_0$ ein Berührpunkt von $D \; \cap \; ]x_0, \infty[$. + Dann hat $f$ in $x_0$ den rechtsseitigen Grenzwert $y_0$ hat, falls + \[ + \lim_{x \to x_0} x \in D \cap ]x_0, \infty[ f(x) = y_0 + .\] Schreibweise + \begin{align*} + &\lim_{x \to x_0^{+}} f(x) = y_0 \\ + &\lim_{x \searrow x_0} f(x) = y_0 + .\end{align*} + \item Sei $x_0$ ein Berührpunkt von $D \cap ]-\infty, x_0[$. Dann hat + $f$ in $x_0$ den linksseitigen Grenzwert $y_0$, falls + \[ + \lim_{x \to x_0} x \in D \cap ]x_0, \infty[ f(x) = y_0 + .\] + Schreibweise: + \begin{align*} + &\lim_{x \to x_0^{-}} f(x) = y_0 \\ + &\lim_{x \nearrow x_0} f(x) = y_0 + .\end{align*} + \end{enumerate} +\end{definition} + +\begin{bsp}[Heaviside Funktion] + $H: \R \to \R$, def. durch + \[ + H(x) := \begin{cases} + 1 & x > 0 \\ + \frac{1}{2} & x = 0 \\ + 0, x < 0 + \end{cases} + .\] Für $x_0 > 0$ gilt $\lim_{x \to x_0} H(x) = 1$. + \begin{proof} + Sei $\epsilon > 0$, wähle $\delta := \frac{x_0}{2} > 0$. Dann gilt + \[ + |H(x) - 1| = 0 < \epsilon \quad \forall x \in \R \; \cap \; ]x_0 - \frac{x_0}{2}, x_0 + \frac{x_0}{2}[ + .\] + \end{proof} + Analog finden wir, dass für $x_0 < 0$ gilt $\lim_{x \to x_0} H(x) = 0$. + + $\lim_{x \to 0} H(x) $ existiert nicht! + \begin{proof} + Angenommen: $\lim_{x \to 0} H(x) = y_0$. + + Dann wählen wir $\epsilon = \frac{1}{4}$, dann ex. $\delta > 0$ mit + \[ + |H(x) - y_0| < \frac{1}{4} \quad \forall x \text{ mit } x \in \;]-\delta, \delta[ + .\] $\implies 1 = |H(-\delta)| - H(\delta)| \le |H(-\delta) - y_0| + |y_0 - H(\delta)| < \frac{1}{4} + \frac{1}{4} = \frac{1}{2}$ Widerspruch! \\ + $\implies \lim_{x \to 0} H(x) $ existiert nicht. + \end{proof} + + Es gibt $\lim_{x \nearrow 0} H(x) = 0$ und $\lim_{x \searrow 0} H(x) = 1$, weil + \begin{align*} + &|H(x) - 0| = 0 \quad \forall x \in \; ]-\delta, 0[ \\ + &|H(x) - 1| = 0 \quad \forall x \in \; ]0, \delta[ + .\end{align*} +\end{bsp} + +\begin{lemma}[Restgliedabschätzung der Exponentialreihe] + \[ + \exp(x) = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = \sum_{n=0}^{N} \frac{x^{n}}{n!} + R_{N+1}(x) + .\], d.h. + \[ + R_{n+1}(x) := \sum_{n=N + 1}^{\infty} \frac{x^{n}}{n!} + .\] Für $R_{n+1}(x)$ gilt + \[ + |R_{n+1}(x)| \le 2 \frac{|x|^{N+1}}{(N+1)!}, \quad \forall |x| \le \frac{N+2}{2}, N \in \N_0 + .\] +\end{lemma} + +\begin{proof} + \begin{align*} + |R_{n+1}(x)| = \left| \sum_{n=N+1}^{\infty} \frac{x^{n}}{n!} \right| + &= \frac{|x|^{N+1}}{(N+1)!} \left( 1 + \frac{|x|}{N+2} + \frac{|x|^2}{(N+2)(N+3)} + \ldots \right) \\ + &\le \frac{|x|^{N+1}}{(N+1)!} \left( 1 + \frac{|x|}{N+2} + \left( \frac{|x|}{N+2} \right)^2 + + \left( \frac{|x|}{N+2} \right)^{3} + \ldots \right) \\ + &\le \frac{|x|^{n+1}}{(N+1)!} \left( 1 + \frac{1}{2} + \frac{1}{4} + \ldots \right) \\ + &= \frac{|x|^{N+1}}{(N+1)!} \sum_{k=0}^{\infty} \left(\frac{1}{2}\right)^{k} \\ + &= 2 \frac{|x|^{N+1}}{(N+1)!} + .\end{align*} +\end{proof} + +\begin{bsp} + \[ + \lim_{x \to 0} \exp(x) = 1 + .\] + \begin{proof} + Sei $\epsilon > 0$, wähle $\delta := \frac{\epsilon}{4}$. Dann gilt $\forall x \in \; ]-\frac{\epsilon}{4}, \frac{\epsilon}{4}[$, wobei + O.B.d.A. $\frac{\epsilon}{4} < 1$, dass + \[ + |\exp(x) - 1| = |R_{0+1}| \le 2 \cdot \frac{|x|^{0+1}}{(0+1)!} = 2 |x| < 2\cdot \frac{\epsilon}{4} = \frac{\epsilon}{2} < \epsilon + .\] + \end{proof} +\end{bsp} + +\begin{lemma}[Folgenkriterium] + Sei $f\colon D \to \R$, $D \subset \R$ und $x_0$ ein Berührpunkt von $D$. Dann gilt + \[ + \lim_{x \to x_0} f(x) = y_0 \quad \iff \quad \forall \text{ Folgen } (x_n)_{n\in\N} \subset D \text{ mit } \lim_{n \to \infty} x_n = x_0 + \text{ gilt } \lim_{n \to \infty} f(x_n) = y_0 + .\] +\end{lemma} + +\begin{proof} + \begin{itemize} + \item ,,$\implies$'': Sei $(x_n)_{n\in\N} \subset D$ mit $\lim_{n \to \infty} x_n = x_0$. + + Zu zeigen: $\lim_{n \to \infty} f(x_0) = y_0$. Sei also $\epsilon > 0$, nach Def. + von $\lim_{x \to x_0} f(x)$ existiert ein $\delta > 0$, s.d. + \[ + |f(x) - y_0| < \epsilon \quad \forall x \in D \text{ mit } |x - x_0| < \delta + .\] Zu $\delta > 0$ ex. ein Index $n_{\delta} \in \N$ mit + $|x_n - x_0| < \delta \quad \forall n \ge n_{\delta}$. + + $\implies |f(x_n) - y_0| < \epsilon \quad \forall n \ge n_\delta$ \\ + $\implies \lim_{n \to \infty} f(x_n) = y_0$ + \item ,,$\impliedby$ '' Zu zeigen.: $\lim_{x \to x_0} f(x) = y_0$, d.h. + \[ + \forall \epsilon > 0 \; \exists \delta > 0\colon |f(x) - y_0| < \epsilon \quad \forall x \in D \colon |x - x_0| < \delta + .\] Angenommen das gilt nicht. + + Dann $\exists \epsilon_0 > 0$, s.d. $\forall \delta > 0$ ein $x \in D$ mit $|x - x_0| < \delta$ und $|f(x)-y_0| \ge \epsilon_0$. \\ + $\implies$ Für alle $n \in \N$ $\exists x_n \in D$ mit $|x_n - x_0| < \frac{1}{n}$ und $|f(x) - y_0| \ge \epsilon_0$ \\ + $\implies$ Diese $(x_n)_{n\in\N}$ definieren eine Folge mit $\lim_{n \to \infty} x_n = x_0$, + aber $|f(x_n) - y_0| \ge \epsilon_0 \quad \forall n \in \N$ \\ + $\implies f(x_n)$ konvergiert nicht gegen $y_0$. Widerspruch!\\ + $\implies$ Annahme ist falsch $\implies$ Behauptung + \end{itemize} +\end{proof} + +\subsection{Stetigkeit} + +\begin{definition}[Stetigkeit] + Sei $f: D \to \R$, und $a \in D$. $f$ heißt stetig im Punkt $a$, falls + \[ + \lim_{x \to a} f(x) = f(a) + .\] $f$ heißt stetig in $D$, falls $f$ stetig in $a$ ist $\forall a \in D$. +\end{definition} + +Äquivalente Definitionen + +\begin{definition}[Stetigkeit per $\epsilon$ / $\delta$ Argument] + $f$ ist stetig in $a$ $\iff$ + \[ + \forall \epsilon > 0 \; \exists \delta > 0 \text{ mit } |f(x) - f(a)| < \epsilon \quad \forall x \in D \text{ mit } |x-a| < \delta + .\] +\end{definition} + +\begin{definition}[Stetigkeit mit Folgen] + $f$ ist stetig in $a$ $\iff$ + \[ + \forall \text{ Folgen } (x_n)_{n\in\N} \text{ in } D \text{ mit } \lim_{n \to \infty} x_n = a \text{ gilt, dass } + \lim_{n \to \infty} f(x_n) = f(a) + .\] +\end{definition} + +\begin{definition}[Stetigkeit mit Bild] + $f$ ist stetig in $a$ $\iff$ + $\forall \epsilon > 0 \; \exists \delta > 0 \text{ s.d. }$ + \[ + f(U_{\delta}(a)) \subset \; ] f(a) - \epsilon, f(a) + \epsilon [ \; = U_{\epsilon}(f(a)) + .\] +\end{definition} + +\begin{bsp} + \begin{enumerate}[(1)] + \item Konstante Funktionen und die Identität sind auf ganz $\R$ stetig. + + Konstante Fkt.: Wähle $\delta$ beliebig, da + \[ + \forall x \in \R\colon |x - a| < \delta \implies 0 = |f(x) - f(a)| < \epsilon + .\] Bei der Identität: Wähle $\delta := \epsilon > 0$, denn + \[ + \forall x \in \R \text{ mit } |x - a| < \delta = \epsilon \implies |f(x)- f(a)| = |x-a| < \epsilon + .\] + \item $|\cdot |: \R \to \R$ ist stetig auf $\R$. Das folgt aus Rechenregeln für Folgen, + \[ + f(x_n) \to f(a), n \to \infty \implies |f(x_n)| \to |f(a)| + .\] + \item $\exp\colon \R \to \R$ ist stetig auf ganz $\R$. + + Sei $a \in \R$. Sei $(x_n)_{n\in\N}$ Folge mit + \[ + \lim_{n \to \infty} x_n = a \implies \lim_{n \to \infty} (x_n - a ) = 0 + .\] Aus $\lim_{x \to 0} \exp(x) = 1$ folgt $\lim_{n \to \infty} \exp(x_n -a) = 1$ + \[ + \implies \lim_{x \to a} \exp(x_n) = \lim_{x_n \to a} \left( \exp(a) + \exp(n-a)) \right) = \exp(a) \cdot 1 = \exp(a) + .\] + \end{enumerate} +\end{bsp} + +\end{document}