Suppose $f$ is a harmonic function on a connected open set $\Omega$ in the complex plane, and suppose also that $f$ is holomorphic on some open subset $U$ of $\Omega$. Prove that $f$ is holomorphic on all of $\Omega$.
Unfortunately, I do not know how to start this problem. There are only a few ways I know how to show a function is holomorphic - using the "limit definition", Morera's Theorem, showing there is some sequence $f_n\subset H(\Omega)$ converging normally to $f$, etc. However, I can't see how to solve this problem with these methods. Could I receive some suggestions on what to be thinking about? Thanks.