A harmonic function on a domain $D$ is infinitely differentiable on $D$

Question

I have the following exercise

Let $u(x,y)$ be a harmonic function on a domain $D$. Prove that u is infinitely differentiable.

Note: In the context of this question, a domain means an open set that is path-connected.

My question is whether the claim in the exercise is true without further requiring that $D$ should be simply connected.

If I knew that $D$ is simply connected, I could find a conjugate harmonic function $v(x,y)$ such that $f=u+iv$ would be holomorphic on $D$ and thus infinitely differentiable and thus $u$ would be infinitely differentiable. But I'm not sure if this claim is still true without requiring that $D$ is simply connected.

Answer 1

$\log|z|$ is harmonic on $\mathbb{C}-\{0\}$ but there doesn't exist any analytic function whose real or imaginary part is $\log|z|$. Therefore, simple connectedness is required.