Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.