places of holomorphic function

If $c$ is a complex constant and $f$ a holomorphic function in a domain $D$ of $ℂ$, then $f$ has in every compact (closed (http://planetmath.org/TopologyOfTheComplexPlane) and bounded (http://planetmath.org/Bounded)) subdomain of $D$ at most a finite set of http://planetmath.org/node/9084$\mathrm{c}$-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 subdomain of $D$.  Suppose that there is an infinite amount of $c$-places of $f$ in $A$.  By http://planetmath.org/node/2125Bolzano–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.