Let $X$ and $Y$ be simply connected open regions of $\mathbb{C}$, and let $Z \subset X$ be a Cantor set. Assume we have a homeomorphism $f$ from $X$ to $Y$, which is holomorphic on $X \setminus Z$. Is $f$ necessarily holomorphic on $X$? 



Functions holomorphic on a region minus a Cantor set
This belongs to the subject of holomorphic removability. See [this Wiki article](http://en.wikipedia.org/wiki/Analytic_capacity#Removable_sets_and_Painlev.C3.A9.27s_problem) for more references. In particular, the article implies that any set with Hausdorff dimension smaller than $1$ is holomorphically removable, and if its Hausdroff dimension is greater than $1,$ it is not. If it is equal to $1,$ you remain puzzled.


