Casorati-Weierstrass theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

Suppose $U\subset \mathbb{C}$ is an open subset with $z_0\in U$, and $f:U\setminus \{z_0\}\to \mathbb{C}$ is a holomorphic function. Then, the following are equivalent:

1. $f$ has an essential singularity at $z_0$
2. For any open neighborhood $V\ni z_0$, the image of $V\setminus \{z_0\}$ under $f$ is dense in $\mathbb{C}$