Let $U \subset \mathbb{C}$ be a connected [open set](/page/Open%20Set), and let $f: U \to \mathbb{C}$ be analytic. If $f$ is not identically zero on $U$, then every zero of $f$ is isolated in $U$; explicitly, for every $a \in U$ such that $f(a)=0$, there exists $r>0$ such that $B(a,r) \subset U$ and
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\begin{align*}
f(z) \neq 0 \quad \text{for every } z \in B(a,r) \setminus \{a\}.
\end{align*}