Q:

Harmonic functions

Let $f: \Omega \to \mathbb{R}$ be a harmonic function, where $\Omega \subset \mathbb{R}^2$ is an open subset. 

What can be said about the points where $$\frac{\partial f}{\partial x} =\frac{\partial f}{\partial y}=0? $$(Is it discrete, empty, or on the boundary of $\Omega$?).

Thanks

A:

The question was settled in the comments: since the $\partial/\partial z$ derivative of a harmonic function is holomorphic, it follows that the zero set is discrete. It may or may not be empty. It does not make much sense to talk about it being on the boundary since $f$ is not assumed to be defined there, let alone differentiable.

