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 $\mathbb {C}$