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)?

Related: Does factor-wise continuity imply continuity?

  • 2
    $\begingroup$ Something that would suffice (but is rather trivial) is if $f(x, \cdot)$ is continuous for every $x$, where the "continuity is uniform w.r.t. $x$", i.e. for $\epsilon > 0$ and $y \in Y$, there is $\delta > 0$ such that $|f(x, y) - f(x,z)| < \epsilon$ holds for all $x$, as soon as $|y-z| < \delta$. $\endgroup$
    – PhoemueX
    Feb 5, 2015 at 17:13
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    $\begingroup$ I'm also very interested to find an answer to this question. Perhaps the OP found some criteria for joint continuity everywhere in the past years? $\endgroup$
    – Sobolev
    Jan 28, 2019 at 19:47


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