It is a basic fact that the zero set of a non zero holomorphic function defined on a open set $A$ is discrete.
By a result in Rudin's textbook on "Real and Complex Analysis", we know that any real valued harmonic function $u$ is locally the real part of some holomorphic function, so I ask the following question:
Is the zero set $Z(u)$ of a non zero real valued harmonic function $u$ defined on open set $A\subset \mathbb{C}$ discrete?
Note that if $u$ vanishes on a nonempty open set $O\subset A$, then write $f=u+iv$ to be the holomorphic function, then Cauchy-Riemman's theorem implies that $v=0$ on $O$, so $f=0$ on $O$, then $f=0$ on $A$.
No. The function $f(a+ib)=b$ is harmonic but its zero set is the real line.