Continuity on a product space, one of which is compactly generated Let $A$ and $B$ be Hausdorff topological spaces, with $A$ compactly generated (i.e., a subset of $A$ is open iff its intersection with each compact subset $K$ of $A$ is open in $K$; from 2nd edition of Munkres, Topology, p. 283).
If function $f:A \times B \rightarrow \mathbb{R}$ is continuous on $K \times B$ for each compact subset $K$ of $A$, is $f$ necessarily continuous on its entire domain? (Product sets are given the product topology)
What I tried
My guess is that the answer is negative. For sequential spaces, the answer is positive: if sequence $(a_n,b_n)$ converges to $(a,b)$, then $K = \{a_n: n \in \mathbb{N}\} \times \{a\}$ is compact, so the sequence belongs to $K \times B$ and by continuity on this set, $f(a,b) = \lim f(a_n,b_n)$. This argument does not extend to nets, so I believe the claim has to be wrong, but I don't see how to make a counterexample.
(Updated) If $A$ is locally compact, the answer seems to be positive as well: since every $a \in A$ has a compact neighborhood $K$, every $(a,b) \in A \times B$ belongs to the interior of $K \times B$ for some compact $K$ and since $f$ by assumption is continuous on this set and consequently on a neighborhood of $(a,b)$, the function has to be continuous at $(a,b)$.
 A: Such a space cannot be compactly generated. If $C\cap K×B$ is closed in $K×B$ for each compact $K\subseteq B,$ then it is closed in each compact subspace of $A×B$, as every compact subspace is a subset of some $K×B$. So if $A×B$ is compactly generated, then the property $(*)$ you describe is satisfied.
On the other hand, if $B$ is compactly generated, then $(*)$ is equivalent to $A×B$ being cg. Note that the product of two compactly generated spaces, one of which is locally compact, is again compactly generated. So if $B$ is cg, then $K×B$ is cg, too. Now if $C∩D$ is closed in each compact $D$, then so it is in each compact $D\subseteq K×B$, so $C∩K×B$ is closed in $K×B$, and then by $(*)$ it closed in the entire product $A×B$.
That means it suffices to find a pair $A,B$ of compactly generated Hausdorff spaces such that $A×B$ is not compactly generated. On such example is where $A=S×I/S×\{0\}$ where $S$ is an uncountable discrete space, and $B=\Bbb N×I/\Bbb N×\{0\}$, so $A$ and $B$ are CW complexes. One can show that $A×B$ is not a CW complex, hence not compactly generated.
