Consider a function $f:\mathcal{X}\times \mathcal{Y}\rightarrow \mathbb{R}$ with $\mathcal{X}\subseteq \mathbb{R}^k$ and $\mathcal{Y}\subseteq \mathbb{R}^p$.
Under which sets of conditions is $\sup_{x\in \mathcal{X}} f(x,y)$ continuous?
Similar questions are asked here and here (among the others) but I can't summarise the main findings.
In particular, is having $f(x,y)$ jointly continuous in $x$ and $y$ plus $\mathcal{X}$ compact sufficient?