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+\documentclass{../../../lecture} + +\begin{document} +\begin{proof} + Sei $f(a) < y < f(b)$ (die Fälle $y = f(a)$ oder $y = f(b)$ sind trivial). + Betrachte $g(x) := f(x) - y$. $g(x)$ stetig und + $g(a) < 0, g(b) > 0$. + + Wir suchen die Nullstelle $c \in [a,b]$ mit $g(c) = 0$ mit dem + Intervallschachtelungsprinzip in $\R$. + + Starte mit $I_0 = [a_0, b_0] := [a,b]$, es gilt + $g(a_0)\cdot g(b_0) < 0$. Sei $c_0 := \frac{1}{2}(a_0+b_0)$ der + Mittelpunkt von $[a_0, b_0]$. Falls $g(c_0) = 0$, dann + ist $c_0$ Nullstelle von $g(x)$. Sonst setze + \[ + I_1 = [a_1, b_1] = \begin{cases} + [a_0, c_0] & \text{für } g(a_0)g(c_0) < 0 \\ + [c_0, b_0] & \text{für } g(c_0)g(b_0) < 0 + \end{cases} + .\] Es gilt $g(a_1) \cdot g(b_1) < 0$ und $|a_1 - b_1| = \frac{1}{2} |a_0-b_0|$ usw. + + Nach endlich vielen Schritten erhalten wir entweder eine Nullstelle $c_n$ von $g(x)$. Dann + ist $c = c_n$, oder eine unendliche Folge von geschachtelten Intervallen $I_n = [a_n, b_n]$, $n \in \N$ mit den + Eigenschaften $g(a_n)g(b_n) < 0$ und + \[ + |b_n - a_n| = \frac{1}{2}|b_{n-1} - a_{n-1}| = \ldots = \left( \frac{1}{2} \right) ^{n} |b_0 - a_0| + .\] wird konstruiert. $\implies$ + \[ + \exists c = \bigcap_{n = 1}^{\infty} I_n \text{ und } c = \lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n + .\] Nach Konstruktion $g(a_n) g(b_n) < 0$. Wegen der Stetigkeit + und den Eigenschaften des Limes gilt $g(a_n)g(b_n) \to g(c)g(c) \le 0, n \to \infty$\\ + $\implies g(c) = 0$ +\end{proof} + +\begin{bem} + \begin{enumerate} + \item Bisektionsverfahren zur Berechnung einer Nullstelle einer stetigen Funktion funktioniert + wie im Beweis des Zwischenwertsatzes. + \item Eine stetige Funktion $f\colon [a,b] \to \R$ mit Bildbereich + $B \subset [a,b]$ besitzt einen ,,Fixpunkt'', d.h. + $\exists c \in [a,b]$ mit $f(c) = c$ (Folgt aus Beweis des + Zwischenwertsatzes mit $g(x) = f(x) - x$ + \item Sei $I \subset \R$ ein Intervall und $f\colon I \to \R$ stetig, dann + ist $f(I)$ ebenfalls ein Intervall. Konvention: $f \equiv a$ konstant, dann $f(I) = [a,a]$. + \begin{proof} + Setze $B := \text{sup}\{f(x) \mid x \in I\} $ falls $f$ nach oben + beschränkt, sonst $B := \infty$ und $A := \text{inf}\{f(x) \mid x \in I\} $ falls + $f$ nach unten beschränkt, sonst $A := -\infty$. Sei $y \in \R$ mit + $A < y < B$. Nach Definition $\exists x_0, x_1 \in I$ mit $f(x_0) < y < f(x_1)$.\\ + $\stackrel{\text{ZWS}}{\implies}$ $\exists x \in I$ mit $f(x) = y$ \\ + $\implies \; ]A,B[ \; \subset f(I).$ Damit: + \[ + f(I) \in \{ ]A, B[, [A, B[, ]A, B], [A, B]\} + .\] + \end{proof} + \end{enumerate} +\end{bem} + +\begin{definition}[Monotone Funktionen] + Sei $D \subset \R$, $f\colon D \to \R$. + \[ + f \text{ heißt } \begin{cases} + \text{monoton wachsend} & f(x) \le f(x') \\ + \text{streng monoton wachsend} & f(x) < f(x') \\ + \text{monoton fallend} & f(x) \ge f(x') \\ + \text{streng monoton fallend} & f(x) > f(x') \\ + \end{cases} + \quad \forall x, x' \in D \text{ mit } x < x' + .\] +\end{definition} + +\begin{satz}[Stetigkeit der Umkehrfunktion] + Sei $I \subset \R$ ein Intervall und $f\colon I \to \R$ eine stetige + streng monoton wachsende (streng monoton fallende) Funktion. + + Dann ist $f\colon I \to f(I)$ bijektiv und $f^{-1}\colon f(I) \to I$ ebenfalls + stetig und streng monoton wachsend (bzw. fallend). +\end{satz} + +\begin{proof} + $f(I)$ ist wieder ein Intervall, $f$ ist streng monoton $\implies$ injektiv + $\implies f:I \to f(I)$ bijektiv, d.h. $\exists f^{-1}$. + + Außerdem $f(x_1) < f(x_2) \implies$ + \[ + \begin{cases} + f^{-1}(f(x_1)) = x_1 < x_2 = f^{-1}(f^{(x_2}) & f \text{ wachsend} \\ + f^{-1}(f(x_1)) = x_1 > x_2 = f^{-1}(f^{(x_2}) & f \text{ fallend} \\ + \end{cases} +.\] $\implies f^{-1}$ auch streng monoton wachsend (bzw. fallend). + + Zu zeigen: $f^{-1}\colon f(I) \to I$ ist stetig. O.B.d.A. $f$ wachsend (sonst $\to -f$ ). + Sei $y_0 \in f(I)$ mit $x_0 := f^{-1}(y_0)$ und $\epsilon > 0$. + + 1. Fall: $x_0$ ist kein Randpunkt, sei $\epsilon > 0$ so klein, dass $]x_0 - \epsilon, x_0 + \epsilon[ \; \subset I$. + Dann + \[ + y_{-} := f(x_0 - \epsilon) < y_0 < f(x_0 + \epsilon) =: y_{+} + .\] Definiere $\delta := \text{min}(y_{+}-y_0, y_0 - y_{-})$. $\implies$ + \[ + .\] + \begin{align*} + &B_{\delta}(y_0) \subset \; ]y_{-}, y_{+}[ \; \stackrel{\text{ZWS}}{\subset} f \; ]x_0 - \epsilon, x_0 + \epsilon[ \qquad \mid f^{-1} + \implies & f^{-1}(]y_0 - \delta, y_0 + \delta[) \subset f^{-1}(f(]x_0 - \epsilon, x_0 + \epsilon[)) = ]x_0 - \epsilon, x_0 + \epsilon[ \\ + &\implies f^{-1} \text{ stetig in } y_0 \text{ nach Definition} + .\end{align*} + + 2. Fall: $x_0$ ist Randpunkt $\implies y_0$ ist Randpunkt. Gleiche Argumentation wie + oben mit $[x_0 - \epsilon, x_0]$ bzw. $[x_0, x_0 + \epsilon]$ +\end{proof} + +\begin{bsp} + \begin{enumerate} + \item Wurzeln sind stetig + + Für $k \in \N$ ist die Funktion $f: [0, \infty[ \to [0, \infty[$, $f(x) := x^{k}$ + streng monoton wachsend und surjektiv.\\ + $\implies$ Die Umkehrfunktion $f^{-1}\colon [0, \infty[ \to [0, \infty[$ ist + stetig und streng monoton wachsend mit $f^{-1}(x) = \sqrt[k]{x} $ + \item $\ln$ ist stetig + \end{enumerate} +\end{bsp} + +\begin{satz}[Logarithmus] + $\exp\colon \R \to \R$, $x \mapsto e^{x}$ ist streng monoton wachsend mit + $\exp(\R) = \; ]0, \infty[$. Die Umkehrfunktion + $\ln\colon ]0, \infty[ \to \R$ ist stetig, streng monoton wachsend und heißt + natürlicher Logarithmus. $\ln(x) = \log_e(x)$. + + Es gibt die Funktionalgleichung + \[ + \ln(x\cdot y) = \ln x + \ln y \quad \forall x, y \in ]0, \infty[ + .\] +\end{satz} + +\begin{proof} + $f(x) = \exp(x) = e^{x} \stackrel{\text{Def.}}{=} \sum_{k=0}^{\infty} \frac{x^{k}}{k!}$. + + \begin{enumerate} + \item $e^{x}$ ist streng monoton wachsend, weil für $k > 0$ gilt $e^{k} > 1$ und + für $x < x'$ folgt $\exists h > 0$ s.d. $x' = x + h$.\\ + $\implies e^{x} < e^{x} \cdot \underbrace{e^{h}}_{> 1} = e^{x'} \implies e^{x} $ injektiv + \item $e^{x}$ surjektiv, weil: Sei $a \in \R$ beliebig. Folge $(e^{n})_{n \in \N}$ divergiert + strikt, da $e > 1$ $\implies$ Folge $(e^{-n})_{n \in \N}$ ist eine Nullfolge.\\ + $\implies \exists n \in \N$ mit $e^{-n} < a < e^{n}$. + + Die Exponentialfunktion ist auf $\R$ und auch auf $[-n, n]$ stetig $\stackrel{\text{ZWS}}{\implies}$ $\exists c \in [-n, n]$, s.d. + $e^{c} = a \implies e^{x}$ surjektiv + \item Nach Umkehrfunktionssatz folgt $\ln(x)$ ist stetig und + strikt monoton wachsend $\forall [e^{-n}, e^{n}]$ $\forall n \in \N$ \\ + $\implies \ln\colon ]0, \infty[ \to \R$ stetig und streng monoton + wachsend. + \item Z.z.: $\ln(x\cdot y) = \ln x + \ln y$. + + Für $x, y$ gilt + \begin{align*} + &\exp(\ln x + \ln y) = e^{\ln x} \cdot e^{\ln y} = x \cdot y \qquad \mid \ln \\ + \implies & \ln(e^{\ln x + \ln y }) = \ln x + \ln y = \ln (x\cdot y) + .\end{align*} + \end{enumerate} +\end{proof} + +\end{document}