Let $f: D - K \rightarrow \mathbb{C}$ be holomorphic, where $D$ is a planar domain and $K$ is a compact subset of $D$. Suppose that $f$ extends continuously to all of $D$. On which conditions on $K$ can we conclude that $f$ is actually holomorphic on all of $D$ ?
Note : I ask that $f$ is continuous and not merely bounded on $D$ ; all the references I have found seem concerned with the bounded case. Then the key concept is analytic capacity, and there is a discussion involving the Hausdorff dimension of $K$. But I heard that there are stronger results if we assume continuity instead of just boundedness. Anyway, using Morera's theorem you can see that this holds for say, smooth curves, for which are not removable if you consider only bounded functions.
If someone has a reference, I would be very glad.