Suppose \(f\not\equiv 0\) is holomorphic on \(\Omega\smts{z_0}\) with a sequence of zeros \(z_k\) limiting to \(z_0\). Show that \(z_0\) is an essential singularity of \(f\).
Solution
It can not be a pole: otherwise \(\abs{f(z)}\to \infty\) as \(z\to z_0\), but \(f(z) = 0\) infinitely often in every neighborhood of \(z_0\) since the \(z_k\) accumulate on it. It can not be removable: otherwise \(f\) extends holomorphically over \(z_0\), and continuity forces \(f(z_k) \to 0\) as \(z_k\to z_0\). But then \(f = 0\) on a set with an accumulation point, making \(f \equiv 0\) by the identity principle.