I want to ask a question of the maximal natural domain of an analytic function. We know that on $\mathbb{C}^n$, a domain $U$ is a domain of holomorphy if and only if $U$ is a pseudoconvex domain, and in particular in complex plane, we have that every open set is pseudoconvex.
Here I want to ask for the converse for $\mathbb{C}$:
Q: For an analytic function $f$, is the maximal set $W$ for which $f$ can be extended necessarily open? And moreover, does this generalize in $\mathbb{C}^n$?
Notice that for some particular cases, this does not raise a question. For instance, if $f$ is defined on some closed set $W$ with $C^1$ boundary(say, a Jordan curve), then an application of Caratheodory theorem and Schwartz reflection could be enough.
It might be a stupid question, but it seems that I could not find a counterexample or a proof yet.