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.

  • $\begingroup$ Most people here would appreciate the question being asked with some motivation and in a less direct way. Moreover, it would be good if you could give some idea of whether you think such an example exists, and what you have already attempted to show this. $\endgroup$ – Joe Tait Jun 11 '14 at 11:29
  • $\begingroup$ I am working on a publishable theorem concerning separate continuity. I am completely at a loss as to the existence of counterexamples. $\endgroup$ – user156495 Jun 11 '14 at 11:34
  • $\begingroup$ Thanks for editing the question. It's now a lot more clear. You may want to consider (if this question does not receive any attention within a week or so) posting a similar question to www.mathoverflow.net which is more geared towards research level questions (research level questions are of course welcome here too, and many people browse both sites). $\endgroup$ – Dan Rust Jun 11 '14 at 14:38

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