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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.

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By separately continuous do you mean that when fixing $x$, the function $y\mapsto f(x,y)$ is continuous from $Y$ to $Z$, and similarly for fixing $y$? –  Asaf Karagila Apr 13 '13 at 11:07
@AsafKaragila: Yes. –  spin Apr 13 '13 at 11:09
What kind of spaces do you want to consider? No such $f$ exists if $X,Y,Z$ are metrizable. –  Martin Apr 13 '13 at 12:20
There should be a lot of such examples in the papers by V. K. Maslyuchenko and his son, O. V. Maslyuchenko. They both are powerful mathematicians and have a family “tradition” or, even, a school, devoted to different weak continuity of maps on the products of different topological spaces. :-) So I suggest you to search and to look their papers devoted to separately continuous not (joint) continuous maps. –  Alex Ravsky Apr 13 '13 at 12:43
@spin: If any spaces can satisfy you, you may take $X=Y=Z=\mathbb{R}$ in the cofinite topology (that is, a nonempty set $A\subset\mathbb{R}$ is open iff $\mathbb{R}\backslash A$ is finite) and $f(x,y)=x+y$ for each $x,y\in\mathbb{R}$. :-) –  Alex Ravsky Apr 14 '13 at 15:19

1 Answer 1

up vote 0 down vote accepted

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.

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