Continuous function on compact metric space has minimum and maximum

Question

Let's have continuous function $F$, $B$ - closed ball around some point $x_0$ with radius $\rho$, and some set $J$. $$\max_{t∈J,x∈B(x_0,ρ)}||F(x,t)||$$

We have one theorem which says that continuous function on compact set has minimum and maximum.

In this case, we have function with two variables. Since we have $B$, which is closed set, do we have to take $J$ to be compact (or just closed), so we can talk about maximum of the function $F$?

Answer 1

If we are talking in $\Bbb R$, let say $F$ is define on $D \subset\Bbb R^2$.

So $(x,y)\in D$ if and only if $x\in J \wedge y\in B_{(x_0,p)}$.

For continuous function on compact metric space to has minimum and maximum, you are asking $D$ to be compact, cause $F$ is define on $D$.

$B$ or $J$ to be compact is not enough.