Under what conditions does [for any $x\in X$, $F(x,-): Y\to Z$ is continuous] imply [$F(x,y): X\times Y \to Z$ is continuous]?

Question

Let $X,Y,Z$ be topological spaces and $F(x,y): X\times Y \to Z$ be mapping.

Under what conditions does [for any $x\in X$, $F(x,-): Y\to Z$ is continuous] imply [$F(x,y): X\times Y \to Z$ is continuous]?

I know when X is discrete space, it holds. However when X is $[0,1]$, it does not hold.

Answer 1

The least you need is that $F(-,y)$ is continuous for every $y$, but even that is not enough.

There are some cases in which continuity in each variable implies continuity, e.g. when $X$ and $Y$ are vector spaces and $F$ is a bilinear map or when $F = 0$ on a dense subset of $X\times Y$. Most of these examples use Baire theorem, which is something I think you'd want to check out.

In general it is a difficult question to determine when a coordinate-wise continuous function (also called "separately continuous" or "separate continuity") is continuous, as far as I know there is no simple condition for that. But you can look for example at this question https://mathoverflow.net/questions/440925/separate-continuity-implies-joint-continuity