Continuity proof for compact domain

Solution 1:

You can't prove it, because it's not true. A two-dimensional counterexample: Let $f(x,w)=w$ on the cross-shaped set $\{(x,w): -2\le x,w\le 2\text{ and }\min(|x|,|w|)\le 1\}$. For this clearly continuous function, $$\max_w f(x,w)=\begin{cases}1&-2\le x< 1\\ 2&-1\le x\le 1\\ 1& 1<x\le 2\end{cases}$$ What additional conditions would we need to make this work? Convexity of the domain should do it.

Addendum: As seen in supinf's answer here, convexity isn't enough. I was hoping to avoid the overkill of making the domain a compact Cartesian product, but it looks like we can't do that.

Proof by contradiction: $ S \subseteq \emptyset \rightarrow S= \emptyset $

In how many ways are we able to arrange $k$ identical non-overlapping dominoes on a circle of $2n$ labelled vertices?

Why is $\zeta(s)=\lim_{x\to\infty}\left(\sum_{n\leqslant x} \frac{1}{n^s}-\frac{x^{1-s}}{1-s}\right)$ for $0<s<1$?

Proving if $b^k = a$ and $\text{ord}(a) = n$ then $\text{ord}(b) = kn$.

Evaluate the limit $\displaystyle\lim_{x \to 0^{+}} x \cdot \ln(x)$

Proving that a Galois group $Gal(E/Q)$ is isomorphic to $\mathbb{F}_p^\times$

Solving a recurrence relation with the characteristic polynomial

Does proving the following statement equate to proving the twin prime conjecture?

Proving that $nb \text{ (mod m)}$ reaches all values $\{0 \dots (m-1)\}$ if $n$ and $m$ are relatively prime

Question about basis step of strong induction proof

Does this equation have positive integer solutions?

$0^0$ is undefined, but sometimes defined as $1$?