Riemann_Removable_Singularities_Theorem

Let $U \subset \C$ be a domain, let $z_0 \in U$, and let $f: U \setminus \left\{ {z_0}\right\} \to \C$ be holomorphic. Then the following are equivalent $(1): \quad f$ extends to a holomorphic function $f: U \to \C$. $(2): \quad f$ extends to a continuous function $f: U \to \C$. $(3): \quad f$ is bounded in a neighborhood of $z_0$. $(4): \quad f \left({z}\right) = o \left({\dfrac 1 {\left|{z - z_0}\right|} }\right)$ as $z \to z_0$.