Consider a function $f : X\times Y \to Z$ where all spaces are compact metric spaces. Assume further that $f_x: y \mapsto f(x,y)$ and $f_y: x \mapsto f(x,y)$ are continuous. I am looking for conditions on $f$ (or its on its $x$ and $y$ sections) to guarantee that $f$ is continuous in $X\times Y$ in the product topology.
I am aware of the result (PDF) that under conditions (met by the assumptions) on $X,Y,Z$, $f$ is continuous for on a dense $G_\delta$ set of $X\times Y$, and of similar structural assumptions (PDF). However, I am looking for conditions on $f$ to ensure continuity everywhere.
Bonus question: any ideas on what happens if $f$ is a correspondence and we weaken continuity to upper-hemi-continuity (both in the assumption and implication)?