The connection between separate continuity and joined continuity has been studied quite a lot. In particular, one has (as a special case of a far more general Theorem from here) the following:
If $U, V \subset \mathbb{R}^n$ are closed and $f\colon U\times V \to \mathbb{R}$ is separately continuous (i.e. for all $y\in V$ the map $U\ni x \mapsto f(x,y)$ is continuous and for all $x\in U$ the map $V\ni y \mapsto f(x,y)$ is continuous), then there is a set $A\subset U$ which is dense in $U$ and $f$ is continuous in $A\times V$.
Now for my question: Is it known whether the same holds true, if one replaces the word "continuous" with "lower semi-continuous" everywhere? Are there any results in that direction?