I need an example of a real-valued separately continuous function that is discontinuous at each point. This is either a well-known folklore fact or a burning problem in separate versus joint continuity, because I could not find an answer in Zbigniew Piotrowski's topical survey on separate continuity. I looked up a number of various papers by the same author and they are full of positive results but no counterexamples are given.
This is hard because for example if $f\colon X\times Y\to\mathbb R$ is separately continuous and $X$ is metrizable, then $f$ is of first Baire class and consequently has a dense set of continuity points. This means that we are looking for nonmetrizable spaces $X,Y$.
There is a rich literature with theorems which assume some topological properties of $X$ and $Y$ and conclude the existence of joint continuity points, but I cannot find counterexamples that are discontinuous everywhere. It is essential for me that the function should be real-valued because I am working on a theorem about separately continuous real-valued functions.