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.
Let $$\partial_z := \frac{1}{2}\left(\frac{\partial}{\partial x} - i\frac{\partial}{\partial y}\right) \qquad \text{and} \qquad \partial_{\bar{z}} := \frac{1}{2}\left(\frac{\partial}{\partial x} + i\frac{\partial}{\partial y}\right)$$
Since $f$ is harmonic on $\Omega$, $\partial_z \partial_{\bar z}f = 0$ on $\Omega$. Hence $\partial_{\bar{z}} f$ is constant on $\Omega$. Since $f$ is holomorphic on $U$, $\partial_{\bar{z}} f = 0$ on $U$. Since $U$ has a point of accumulation, $\partial_{\bar{z}} f = 0$ on $\Omega$ by the identity theorem for holomorphic functions. Hence, $f$ is holomorphic on $\Omega$.