Continuity proof "If $f(x,y,z,w)$ is continuous and domains of $x,y,z,w$ are all convex set, then $\max_{w} f(x,y,z,w)$ is continuous."

Question

There is the answer about the continuity.

Continuity proof for compact domain

I want to know how to prove that "If $f(x,y,z,w)$ is continuous and domains of $x,y,z,w$ are all convex set, then $\max_{w} f(x,y,z,w)$ is continuous."

Answer 1

Unfortunately, what jmerry conjectured regarding convexity was wrong.

Even if there is a convex compact domain, then $\max_w f(x,y,z,w)$ does not need to be continuous.

For our counterexample, wlog we omit the $z$ variable.

Consider the set $$ C=\{(x,y,w)\in\mathbb R^3 | 0 \leq w \leq 1, (x-w)^2+y^2\leq w^2 \}. $$ Then it can be shown that this set is convex and compact. We define the function $f(x,y,w):=-w$.

Then, for $x,y$ such that $(x-1)^2+y^2=1$ but $x\neq0$ we have $$ g(x,y):=\max_w f(x,y,w)=f(x,y,1)=-1. $$ For $x=y=0$ instead we have $$ g(0,0):=\max_w f(x,y,w)=f(0,0,0)=0. $$ Since there is a sequence $(x_n,y_n)$ such that $(x_n-1)^2+(y_n-1)^2=1$ and $x_n\neq0$ and $(x_n,y_n)\to(0,0)$, this shows that $g$ is not continuous.