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# places of holomorphic function

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.

## Mathematics Subject Classification

30A99*no label found*

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