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