Removable singularities 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

The removable singularities theorem, sometimes termed the Riemann removable singularities theorem, states that if $U$ is a neighborhood of a point $z_{0}$, and $f:U\setminus \{z_{0}\}\to \mathbb {C}$ is a holomorphic function with $\lim _{z\to z_{0}}f(z)$ existing, then $f$ extends to a holomorphic function on $U$, by setting $f(z_{0})=\lim _{z\to z_{0}}f(z)$.