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 $$g(z) \;=\; c$$ 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 $$g(z) \;=\; f(z).$$ Consequently, the identity theorem of holomorphic functions implies that $$f(z) \;=\; g(z) \;=\; c$$ in the whole $D$ . Q.E.D.