Let $u(x,y)$ be a non-constant harmonic function in region $\mathbb{D}_{\mathbb{R}}=: D$ and $$A:=\{(x,y)\in D : u_x = u_y = 0\} $$ what can one say about the set $A$?
Since $u$ is harmonic, there exists a coharmonic (is this the correct term?) function $v$ and $u,v$ satisfy the Cauchy-Riemann equations from which: $$u_x = 0 = v_y\Longrightarrow v=h(x)\quad\mbox{and} -u_y = 0 = v_x\Longrightarrow v=g(y) $$ How do we interpret this? Can a function be only a function of $x$ and only a function of $y$ at the same time? If so, then $v$ is surely constant and that makes $u$ constant. $A=\emptyset$?
..or does it imply that $h,g$ are exactly the same? I'm confused. If they were exactly the same, then by differentiating w.r.t either of $x,y$ we must get $0$. It has to be constant.