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+\begin{aufgabe} + \begin{enumerate}[(a)] + \item Es seien $f_k\colon \R^{3} \setminus M \to \R$ und + $M \coloneqq \{(x,y,z)^{T} \in \R^{3} \mid x = y = 0, z \in \R\} $. + \begin{enumerate}[(i)] + \item Beh.: $f_1(x,y,z) := \frac{xz^2 + y^{3}}{x^2 + y^2}$ ist nicht stetig fortsetzbar + in $(0,0,0)^{T}$. + \begin{proof} + Sei $c \in \R$ beliebig. Dann wähle + \[ + a_n \coloneqq \begin{pmatrix} \frac{1}{n^{3}} \\ 0 \\ \frac{1}{n} \end{pmatrix} + \in \R^{3} \setminus M \text{ es gilt: } a_n \xrightarrow{n \to \infty} 0 + .\] Aber + \begin{align*} + f_1(a_n) = \frac{\frac{1}{n^{3} + \frac{1}{n^2}}}{\frac{1}{n^{6}}} + = \frac{\frac{1}{n^{5}}}{\frac{1}{n^{6}}} = \frac{n^{6}}{n^{5}} = n + \xrightarrow{n \to \infty} \infty > c + .\end{align*} + \end{proof} + \item Beh.: $f_2(x,y,z) \coloneqq \frac{xyz + xy^2}{x^2 + y^2}$ ist mit + $f_2(0, 0, 0) = 0$ stetig fortsetzbar. + \begin{proof} + Sei $\epsilon > 0$ beliebig dann wähle $\delta \coloneqq \frac{\epsilon}{2}$. + Dann folgt $\forall (x,y, z)^{T} \in \mathbb{K}^{n}$ mit + $\left\Vert (x,y,z)^{T} - 0\right\Vert_\infty = \Vert (x,y,z) \Vert_{\infty} < \delta $: + Falls $x \ge z$: + \[ + \left| \frac{xyz + xy^2}{x^2 + y^2} \right| \le \frac{|xyz| + |xy^2|}{x^2 + y^2} + = \frac{|z| + |y|}{\frac{|x|}{|y|} + \frac{|y|}{|x|}} + \le \frac{|z|+ |y|}{\frac{|x| + |y|}{\delta}} + = \frac{\delta (|z| + |y|)}{|x| + |y|} + \le \frac{\delta (|x| + |y|)}{|x| + |y|} = \delta < \epsilon + .\] Falls $y > x$: + + \[ + \left| \frac{xyz + xy^2}{x^2 + y^2} \right| \le \frac{|xyz| + |xy^2|}{x^2 + y^2} + = \frac{|z| + |y|}{\frac{|x|}{|y|} + \frac{|y|}{|x|}} + \le \frac{|z| + |y|}{\underbrace{\frac{|x|}{|y|}}_{\le 1} + 1} + \le |z| + |y| \le 2 \delta = \epsilon + .\] + Falls $z > x$ und $y \le x$: + \[ + \left| \frac{xyz + xy^2}{x^2 + y^2} \right| + \le \frac{|xyz|}{x^2 + y^2} + \frac{|xy^2|}{x^2 + y^2} + \le |z| \frac{x^2}{x^2 + y^2} + |z| \frac{y^2}{x^2 + y^2} + = |z| \le \delta < \epsilon + .\] + Also $f_2$ stetig fortsetzbar in $(0,0,0)^{T}$. + \end{proof} + \end{enumerate} + \item Beh.: Auf dem Produktraum $P \coloneqq \mathbb{K}^{n} \times \mathbb{K}^{n}$ ist + $f\colon P \to \R$ mit + $f(x,y) \coloneqq (x,y)_K$ stetig. Hier ist $(\cdot , \cdot )_K$ Skalarprodukt auf $\mathbb{K}^{n}$ + mit $(x,x)_{K} = \Vert x \Vert_K^2$. + \begin{proof} + Sei $\{x,y\} \in P$ und + $\{x_n, y_n\} \in P$ beliebig mit $\{x_n, y_n\} \xrightarrow{n \to \infty} \{x, y\}$. Damit + folgt: + \[ + \Vert \{x_n, y_n\} - \{x, y\} \Vert_P + = ( \underbrace{\Vert x_n - x \Vert_K^2}_{\ge 0} + + \underbrace{\Vert y_n - y \Vert_K^2}_{\ge 0})^{\frac{1}{2}} + \xrightarrow{n \to \infty} 0 + .\] Damit folgt $\Vert x_n - x \Vert \xrightarrow{ n \to \infty} 0$ und + $\Vert y_n - y \Vert \xrightarrow{ n \to \infty} 0$. Wegen + $(x,x)_K = \Vert x \Vert_K^2$ $\forall x \in \mathbb{K}^{n}$ + gilt die Cauchy-Schwarz-Ungleichung. Damit folgt + \begin{salign*} + | f(x_n, y_n) - f(x,y) | &= | (x_n, y_n)_K - (x,y)_K | \\ + &= | (x_n, y_n)_K - (x_n, y)_K + (x_n, y)_K - (x,y)_K | \\ + &= | (x_n, y_n - y) + (x_n - x, y) | \\ + &\stackrel{\text{C.S.U.}}{\le } + \underbrace{\Vert x_n \Vert_K}_{\xrightarrow{n \to \infty} \Vert x \Vert_K} + \underbrace{\Vert y_n - y \Vert_K}_{\xrightarrow{n \to \infty} 0} + + \Vert y \Vert_K + \underbrace{\Vert x_n - x \Vert_K}_{\xrightarrow{n \to \infty} 0} \\ + &\xrightarrow{n \to \infty} 0 + .\end{salign*} + Damit folgt, dass $f$ stetig ist in $\{x,y\} $. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Beh.: Sei $T \colon \partial K_R(0) \to \R$ stetig. Dann ex. ein $x \in \partial K_R(0)$ + mit $T(x) = T(-x)$. + \begin{proof} + Es ist $\partial K_R(0)$ beschränkt und abgeschlossen, also kompakt. Da $T$ stetig, + existieren also $x_{\text{min}}, x_{\text{max}} \in \partial K_R(0)$ mit + $\displaystyle \max_{x \in \partial K_R(0)} T(x) = T(x_{\text{max}})$ und + $\displaystyle \min_{x \in \partial K_R(0)} T(x) = T(x_{\text{min}})$. + + Definiere + $f\colon \partial K_R(0) \to \R$, $x \mapsto T(x) - T(-x)$. Falls + $f(x_{\text{max}}) = 0$ oder $f(x_{\text{min}}) = 0$, folgt die Behauptung. Sonst + gilt + \[ + f(x_{\text{max}}) \cdot f(x_{\text{min}}) = 0 + .\] Denn falls $T(x_{\text{max}}) > 0$, dann ist + \begin{align*} + f(x_{\text{max}}) &= T(x_{\text{max}}) - T(- x_{max}) > 0 \\ + f(x_{\text{min}}) &= T(x_{\text{min}}) - T(- x_{\text{min}}) < 0 + .\end{align*} + Analog für $T(x_{\text{max}}) < 0$. + Da $\partial K_R(0)$ zusammenhängend und $T$ stetig, folgt mit Zwischenwertsatz, dass + $\exists \xi \in \partial K_R(0)$ mit $f(\xi) = 0 \implies T(\xi) = T(-\xi)$. + \end{proof} + \item Seien $f, g\colon \mathbb{K}^{n} \to \R$ stetig. Definiere für $x \in \mathbb{K}^{n}$: + \[ + \varphi(x) \coloneqq \text{max}(f(x), g(x)) + .\] Beh.: $\varphi\colon \mathbb{K}^{n} \to \R$ ist stetig. + \begin{proof} + Zunächst gilt $\forall a, b \in \R$: + \[ + \max(a,b) = \frac{a+b}{2} + \frac{|a-b|}{2} + ,\] denn: O.E.: $a \ge b$. Dann ist + \[ + \frac{a+b}{2} + \frac{|a-b|}{2} = \frac{a+b}{2} + \frac{a-b}{2} = a = \max(a,b) + .\] + Damit folgt + \[ + \varphi(x) = \max(f(x), g(x)) = \frac{f(x) + g(x)}{2} + \frac{|f(x) - g(x)|}{2} + .\] Da $f, g$ stetig, ist auch $|f-g|$ stetig und da $2 \neq 0$ folgt + $\varphi$ als Summe bzw. Quotient stetiger Funktionen stetig. + \end{proof} + \end{enumerate} +\end{aufgabe} + +\begin{aufgabe} + Sei $f\colon \mathbb{K}^{n} \to D \subseteq \mathbb{K}^{n}$ beliebig + und $g\colon D \subseteq \mathbb{K}^{n} \to \mathbb{K}$ stetig und injektiv, wobei + $D$ kompakt ist. + + Beh.: $g \circ f$ stetig $\implies$ $f$ stetig und $g \circ f$ gleichmäßig stetig + $\implies$ $f$ gleichmäßig stetig. + \begin{proof} + Definiere $B \coloneqq \text{im}(g)$ und $\tilde{g}\colon D \to B$, $x \mapsto g(x)$. + Da $g$ injektiv und stetig, ist auch $\tilde{g}$ injektiv und stetig. + Inbes. ex. $\tilde{g}^{-1}\colon B \to D$ mit $\tilde{g}^{-1}$ stetig. + + Damit gilt $\forall x \in \mathbb{K}^{n}$: $\tilde{g}^{-1}(g(f(x))) = \tilde{g}^{-1}(\tilde{g}(f(x))) + = f(x)$. Also $f = \tilde{g}^{-1} \circ g \circ f$. + + Sei nun $g \circ f$ stetig. Dann folgt + \[ + f = \underbrace{\tilde{g}^{-1}}_{\text{stetig}} \circ \underbrace{g \circ f}_{\text{stetig}} + .\] Also $f$ als Komposition von zwei stetigen Funktionen stetig. + + Sei nun $g \circ f$ gleichmäßig stetig. Da $D$ kompakt und $g$ stetig, ist + $B = \text{im}(g)$ auch kompakt. Also wegen $\tilde{g}^{-1}$ stetig, + $\tilde{g}^{-1}\colon B \to D$ gleichmäßig stetig. Damit folgt + \[ + f = \underbrace{\tilde{g}^{-1}}_{\text{glm. stetig}} \circ \underbrace{g \circ f}_{\text{glm. stetig}} + .\] Also $f$ als Komposition zweier gleichmäßig stetiger Funktionen, gleichmäßig stetig. + \end{proof} +\end{aufgabe} + +\end{document}