I cannot find a proof of this theorem, but I have a strong feeling that it is true. Can someone remind me where to find it (or if I am confused and it is false)?
Let $f: X \times Y \rightarrow \mathbb R^d$ be a continuous function, where $X\subset \mathbb R^n, Y\subset \mathbb R^m$ are compact sets.
Then the functions $g(x) = \sup_y f(x,y)$ and $h(x) = \inf_y f(x,y)$ are also continuous.
Edit: Can you provide a textbook reference, instead of a website? Thanks