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48 lines
2.1 KiB
TeX
48 lines
2.1 KiB
TeX
\chapter{Constant functions}
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\section{Defined on $\mdr$}
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Let $f(x) = c$ with $c \in \mdr$ be a constant function.
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\begin{figure}[htp]
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\centering
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\begin{tikzpicture}
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\begin{axis}[
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legend pos=north west,
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axis x line=middle,
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axis y line=middle,
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grid = major,
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width=0.8\linewidth,
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height=8cm,
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grid style={dashed, gray!30},
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xmin=-5, % start the diagram at this x-coordinate
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xmax= 5, % end the diagram at this x-coordinate
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ymin= 0, % start the diagram at this y-coordinate
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ymax= 3, % end the diagram at this y-coordinate
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axis background/.style={fill=white},
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xlabel=$x$,
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ylabel=$y$,
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tick align=outside,
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minor tick num=-3,
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enlargelimits=true,
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tension=0.08]
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\addplot[domain=-5:5, thick,samples=50, red] {1};
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\addplot[domain=-5:5, thick,samples=50, green] {2};
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\addplot[domain=-5:5, thick,samples=50, blue, densely dotted] {3};
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\addplot[black, mark = *, nodes near coords=$P$,every node near coord/.style={anchor=225}] coordinates {(2, 2)};
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\addplot[blue, mark = *, nodes near coords=$P_{h,\text{min}}$,every node near coord/.style={anchor=225}] coordinates {(2, 3)};
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\addplot[green, mark = x, nodes near coords=$P_{g,\text{min}}$,every node near coord/.style={anchor=120}] coordinates {(2, 2)};
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\addplot[red, mark = *, nodes near coords=$P_{f,\text{min}}$,every node near coord/.style={anchor=225}] coordinates {(2, 1)};
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\draw[thick, dashed] (axis cs:2,0) -- (axis cs:2,3);
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\addlegendentry{$f(x)=1$}
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\addlegendentry{$g(x)=2$}
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\addlegendentry{$h(x)=3$}
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\end{axis}
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\end{tikzpicture}
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\caption{Three constant functions and their points with minimal distance}
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\label{fig:constant-min-distance}
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\end{figure}
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Then $(x_P,f(x_P))$ has
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minimal distance to $P$. Every other point has higher distance.
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See Figure~\ref{fig:constant-min-distance}.
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\section{Defined on a closed interval of $\mdr$}
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