Let me assume that the function $f(x,y,z,w)$ is continuous. Is the $\max_{w} f(x,y,z,w)$ continuous?

Question

Let me assume that the function $f(x,y,z,w)$ is continuous. Is the $\max_{w} f(x,y,z,w)$ continuous?

Since $f(x,y,z,w)$ is continuous, it is seperately continuous for each $x,y,z,$ and $w$.

Answer 1

Hello and welcome to math.stackexchange. This is a nice question.

The answer depends on the domain of the function $f$. If this is not restricted, the answer is No. For example, consider $$f(x,w) = f_0(x \cdot w)$$ for $w \ge 0, \, x \in \mathbb{R}$ where $$ f_0(x) = \begin{cases} -1 \; (x < -1) \\ x \; (-1 \le x \le 1) \\ 1 \; (x > 1) \end{cases} $$

Then $\max_w f(x,w) = 1$ if $x > 0$ and $\max_w f(x,w) =0$ if $x \le 0$. This is not continuous.

Edited

At the very least, the $x$ - sections of the domain of $f$ should be compact sets. As shown in other posts and comments, this is however not sufficient.