A theorem of Hartog states that if $U \subseteq \mathbb C$ is open and $f : U \times U \to \mathbb C$ is analytic when we fix any variable ('separately analytic'), then it is continuous.
In general, it is not true that separate continuity implies joint continuity (See e.g. Functions continuous in each variable: $\frac{xy}{x^2+y^2} : \mathbb R \times \mathbb R \to \mathbb R$ is a counterexample).
One can show that separate continuity implies continuity in (at least) a dense $G_\delta$-set (Theorem 1.2 in this article).
If $X$ is a topological space and $U \subseteq \mathbb C$ open, what can we say about separately continuous functions $f : X \times U \to \mathbb C$ that are analytic in the second variable?
Does this imply that $f$ jointly continuous, or are there counterexamples?
Let's say $X$ is an open subset of $\mathbb R^n$.
What if $f$ is in addition real analytic in the first variable?
I know that:
- $f$ is continuous in open sets where it is locally bounded, by using Cauchy's integral formula.
- In particular, if $X_0 \subseteq X$ and $f$ is continuous on $X_0 \times J$ with $J$ a smooth Jordan curve, then it is continuous on $X_0 \times $ the interior of $J$. Thus isolated discontinuities like the one of $\frac{xy}{x^2+y^2}$ cannot occur.
- An application of the Baire Category Theorem shows that for $K \subseteq X$ compact there exists an open dense subset $V \subseteq U$ such that $f$ is continuous on $K \times V$. Similarly for $L \subseteq U$ compact. See also this question: If a function on a product space is continuous in each variable, is it locally bounded?
We've used holomorphy here, but not enough, because the counterexample $\frac{xy}{x^2+y^2}$ satisfies the conclusions so far.
Note how we recover the dense $G_\delta$-result by letting $K$ grow and taking intersections of the $V$'s, relatively easily under the additional hypothesis of analyticity in the second variable.
I realize that this generalizes Hartog's theorem, so it is either false or hard to prove.
