Let $X,Y$ be compact metric spaces and consider $f:X\times Y\rightarrow X$ a separately continuous function.

I am wondering if there could be some additional conditions on $f$ (for example $f(\cdot,y):X\rightarrow X$ being surjective or injective for every $y\in Y$) which would grant the joint continuity of $f:X\times Y\rightarrow X$.

The strongest result I have found is [contained in this article by Namioka][1] and states that in this case there is a dense subset $A\subset Y$
 such that $f:X\times A\rightarrow X$ is joint continuous, but this is not what I am looking for.

  [1]: https://msp.org/pjm/1974/51-2/pjm-v51-n2-p23-s.pdf