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added functions; improved colors

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Martin Thoma 2012-09-23 16:05:14 +02:00
parent 20ca02e155
commit 33b28743ca
2 changed files with 23 additions and 8 deletions
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12
tikz/stetigkeit-differenzierbarkeit/README
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@ -1,4 +1,5 @@
Interessante Funktionen:
========================
Dirichlet-Funktion
* Überall unstetig
@ -11,4 +12,15 @@ f:(0,1)->R
* Stetig, aber nicht gleichmäßig stetig
* Differenzierbar
Sätze
-----
Jede auf einem kompakten Intervall stetige Funktion
$f: [a, b] \rightarrow \mathbb{R}$ ist dort gleichmäßig stetig.
-- Analysis I, Otto Forster, S. 112 (10. Auflage)
LP-Stetigkeit => Glm. Stetigkeit => Stetigkeit
Differenzierbarkeit => Stetigkeit
ACHTUNG
=======
Die Definitionsbereiche müssen richtig gewählt werden, damit die Aussagen stimmen!

19
tikz/stetigkeit-differenzierbarkeit/stetigkeit-differenzierbarkeit.tex
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@ -53,7 +53,7 @@
\draw[fill=yellow!20,yellow!20, rounded corners] (-1.85, 0.70) rectangle (13.4,-6.85);
\draw[fill=lime!20,lime!20, rounded corners] (-1.75, 0.45) rectangle (7.3,-6.75);
\draw[fill=lime!20,lime!20, rounded corners] (-1.75, 0.60) rectangle (7.3,-6.75);
\draw[fill=purple!20,purple!20, rounded corners] (-1.65,-1.55) rectangle (7.2,-6.65);
\draw[fill=blue!20,blue!20, rounded corners] ( 4.55,-3.45) rectangle (13.1,-6.55);
\draw (0, 0) node[algebraicName] (A) {gleichmäßig stetig}
@ -65,6 +65,7 @@
}
(6, 0) node[example, draw=lime, fill=lime!15] (X) {\tiny$f_5(x)=\sin(x)$}
(6,-1) node[example, draw=lime, fill=lime!15] (X) {\tiny$f_6(x)=\cos(x)$}
(4,-1) node[example, draw=lime, fill=lime!15] (X) {\tiny$f_9(x)=\sqrt x$}
(0,-2) node[algebraicName, purple] (C) {Lipschitz-stetig}
(3.5,-2) node[explanation] (X) {
\begin{minipage}{90\textwidth}
@ -73,11 +74,11 @@ $f$ heißt auf $D$ \textbf{Lipschitz-stetig}\\
$:\Leftrightarrow \exists L\ge 0: |f(x)-f(z)|\le L|x-z|\ \forall x,z \in D$
\end{minipage}
}
(12,-6) node[example, draw=black, fill=black!15] (G) {\tiny$f_2(x) = e^x$}
(12,-6) node[example, draw=blue, fill=black!15] (G) {\tiny$f_2(x) = e^x$}
(0,-6) node[example, draw=red, fill=red!15] (K) {\tiny$f_4(x) = |x|$}
(6,-6) node[example, draw=red, fill=red!15, pattern=north east lines wide, pattern color=black!25] (N) {\tiny$f_7(x) = 42$}
(6,-4) node[example, draw=red, fill=red!15, pattern=north east lines wide, pattern color=black!25] (ANCHORD) {\tiny$f_3(x) = 42$}
(0,-6) node[example, draw=purple, fill=red!15] (K) {\tiny$f_4(x) = |x|$}
(6,-6) node[example, draw=purple, fill=red!15, pattern=north east lines wide, pattern color=black!25] (N) {\tiny$f_7(x) = 42$}
(6,-4) node[example, draw=purple, fill=red!15, pattern=north east lines wide, pattern color=black!25] (ANCHORD) {\tiny$f_3(x) = 42$}
(12,-2) node[example, draw=yellow, fill=yellow!15] (Q) {\tiny$f_1(x) = |x|$}
@ -93,7 +94,8 @@ $:\Leftrightarrow \exists L\ge 0: |f(x)-f(z)|\le L|x-z|\ \forall x,z \in D$
\end{minipage}
}
(12,-4) node[example, draw=black, fill=black!15] (P) {\tiny$g_1(x) = \frac{1}{x}$}
(12,-4) node[example, draw=blue, fill=black!15] (P) {\tiny$g_1(x) = \frac{1}{x}$}
(12,-5) node[example, draw=blue, fill=black!15] (P) {\tiny$f_8(x) = x^2$}
(9, -4) node[algebraicName] (random1) {differenzierbar}
@ -106,14 +108,15 @@ $:\Leftrightarrow \exists L\ge 0: |f(x)-f(z)|\le L|x-z|\ \forall x,z \in D$
\end{minipage}
};
% differenzierbar
\draw[blue, thick, rounded corners] ($(ANCHORD.north west)+(-0.5,0.2)$) rectangle ($(G.south east)+(0.2,-0.2)$);
% LP-Stetig
\draw[purple, thick, rounded corners] ($(C.north west)+(-0.3,0.1)$) rectangle ($(N.south east)+(0.3,-0.3)$);
% gleichmäßig stetig
\draw[lime, thick, rounded corners] ($(A.north west)+(-0.4,0.1)$) rectangle ($(N.south east)+(0.4,-0.4)$);
% stetige funktionen
\draw[yellow, thick, rounded corners] ($(A.north west)+(-0.5,0.2)$) rectangle ($(G.south east)+(0.5,-0.5)$);
% differenzierbar
\draw[blue, thick, rounded corners] ($(ANCHORD.north west)+(-0.5,0.2)$) rectangle ($(G.south east)+(0.2,-0.2)$);
\end{tikzpicture}
\end{preview}
\end{document}