Are max and min of a function continuous?

It seems that we have a continuous function $f:\ C\times C\to{\mathbb R}, \ (x,P)\to f_P(x)$, where $C$ is a compact set in ${\mathbb R}^n$. For a given $P\in C$ one can ask about the value $m(P):=\max\{f_P(x)| x\in C\}$, and the question is whether the new function $m(\cdot):\ P\mapsto m(P)$ is continuous on $C$.

This is indeed the case. Note that $f$ is uniformly continuous on the compact set $C\times C$. This means that given an $\epsilon>0$ there is a $\delta>0$ with $f_{P_0}(x_0)<f_P(x)+\epsilon$ whenever $|x-x_0|<\delta$ and $|P-P_0|<\delta$. For fixed $P_0\in C$ there is a point $x_0\in C$ with $m(P_0)=f_{P_0}(x_0)$, and therefore we have $$m(P_0)< f_P(x_0)+\epsilon\leq m(P)+\epsilon$$ for all $P$ with $|P-P_0|<\delta$. By symmetry it follows that $|m(P)-m(P_0)|<\epsilon$ as soon as $|P-P_0|<\delta$, which proves that the function $m(\cdot)$ is continuous on $C$.