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 $f : U ∖ {z_{0}} \to C$ is a holomorphic function with $lim_{z \to z_{0}} f (z)$ existing, then $f$ extends to a holomorphic function on $U$ , by setting $f (z_{0}) = lim_{z \to z_{0}} f (z)$ .