Let $C$ be the Cantor set and $f$ be a bounded holomorphic function on $\{|z|<2\}\setminus C$. Show that $f$ can be extended to a holomorphic function on $\{|z|<2\}$.
My friend gave me this problem, although I don't know much complex analysis. To show that a function can be extended to a larger domain, you show that the limits of the function at the new points exist. I don't see how to do that. I don't know how to use $f$ being bounded, although I realize it's important because otherwise the statement is false for $f=\tfrac{1}{z-2/3}$, since $\tfrac{2}{3}\in C$. This problem piqued my curiosity, and I'm interested in a solution.