Let's say that we have a function $f : A \times B \to X$. For every $a \in A$ we get a function
$$ f_a : B \to X, \ f_a(b) = f(a, b),$$
and for every $b \in B$ we get $f_b : A \to X$ in similar fashion. It isn't hard to show that continuity of $f: A \times B \to X$ (in product topology) implies continuity of every $f_a, f_b$. But I suspect that converse isn't true, just because I have never seen such theorem. I'll be grateful for some easy examples of such behavior (or a theorem disproving my hunch).
