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Let $\Omega \subset \Bbb{C}$ a region. $f:\Omega \to \Bbb{C}$ holomorphic, $f\neq 0$. Show that the set of zeros of $f$ is discrete. (That is, that it doesn't have any limit points.)

This is the second part of an excercise in which I proved already that given $a \in \Omega$ such that $f(a)=0$ there exist $k \in \Bbb{N}$ such that $f(z)=(z-a)^kg(z)$ with $g:\Omega \to \Bbb{C}$ holomorphic and $g(a)\neq 0$. However Im not being able to see how to show that the set of zeros is discrete. Any hint?

nilo de roock
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Joaquin Liniado
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2 Answers2

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You probably also showed that you can choose $k$ such that $g(a) \neq 0$.

Then it follows by continuity of $g$ that $g(z) \neq 0$ in a neighbourhood of $a$, so the only zero of $f$ in that neighbourhood is at $z=a$.

mrf
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This follows from what is referred to as the Identity Theorem: if the set of points on which $f\equiv a$ has a limit point, then $f $ is constant.