Does factor-wise continuity imply continuity?

Question

Let $f$ denote a map from a product space $X \times Y$ to $Z$. If for every $x\in X$, the map $f(x,-)$ is continuous, and the same holds for every $y \in Y$, then is $f$ continuous in general? If not, is there any condition to be imposed to make $f$ continuous?

Answer 1

No, separate continuity does not in general imply joint continuity. The function

$$f:\Bbb R^2\to\Bbb R:\langle x,y\rangle\mapsto\begin{cases}\frac{xy}{x^2+y^2},&\text{if }\langle x,y\rangle\ne\langle0,0\rangle\\\\0,&\text{if }\langle x,y\rangle=\langle 0,0\rangle\end{cases}$$

is continuous everywhere except at the origin. At the origin it is continuous in each variable separately, but it is not continuous.

Added: Useful keywords on which to search are separate continuity and joint continuity. Z. Piotrowski, a former colleague of mine, did quite a lot of work in this area; you’ll find many of his papers here, and in them both results and further references. Be warned, though, that the links don’t always match the text: the link to the comprehensive (if now dated) survey Separate and joint continuity is actually at number $17$, not at number $19$. As I recall, there are more results giving conditions under which the set of points of continuity contains a dense $G_\delta$-set in $X\times Y$ than there are giving conditions that guarantee that a separately continuous function is jointly continuous.

Added 8 March 2015: Number $51$ is an updated survey.