What are some good examples of separately continuous functions $f: X \times Y \rightarrow Z$ that are discontinuous at every point?
Here's a theorem to rule out some spaces: link for a reference
Theorem: Let $X$ be locally compact or completely metrizable, $Y$ compact Hausdorff, $Z$ a metric space. If $f: X \times Y \rightarrow Z$ is separately continuous, then there exists a dense $G_\delta$ subset $A$ of $X$ such that $f$ is continuous on $A \times Y$.
So no example exists for $X$, $Y$ and $Z$ satisfying the assumptions of the theorem.