Separately continuous functions that are discontinuous at every point

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.

Solution 1:

Answering so that this isn't unanswered. In the comments, Alex Ravsky suggested the following example which was just what I needed.

Consider any infinite group $G$ equipped with the cofinite topology. Then the multiplication map $G \times G \rightarrow G$ defined by $(x,y) \mapsto xy$ is separately continuous everywhere, but is not jointly continuous at any point.