If $c$ is a complex constant and $f$ a holomorphic function in a domain $D$ of $\mathbb{C}$, then $f$ has in every compact (closed and bounded) subdomain of $D$ at most a finite set of $c$-places, i.e. the points $z$ where  $f(z)=c$,  except when  $f(z)\equiv c$  in the whole $D$.

Proof.  Let $A$ be a closed and bounded subdomain of $D$.  Suppose that there is an infinite amount of $c$-places of $f$ in $A$.  By Bolzano–Weierstrass theorem, these $c$-places have an accumulation point $z_{0}$, which belongs to the closed set $A$.  Define the constant function $g$ such that

for all $z$ in $D$.  Then $g$ is holomorphic in the domain $D$ and  $g(z)=c$  in an infinite subset of $D$ with the accumulation point $z_{0}$.  Thus in the $c$-places of $f$ we have

Consequently, the identity theorem of holomorphic functions implies that

in the whole $D$.  Q.E.D.