Finding all Harmonic functions on given domain

Question

let $\Omega$ be a domain in $\mathbb{C}$. find all harmonic functions $U:\Omega \to\mathbb{R}$ such that the set $$ G=\left\{z \in \Omega: \frac{\partial u}{\partial x} =0, \frac{\partial u}{\partial y} =0\right\}$$ has a limit point in $\Omega$.

how to approach this.

Answer 1

Hint: $u_x-iu_y$ is holomorphic in $\Omega.$