Holomorphic function is determined by its germ

This fact is an application of the following pivotal fact/result/idea: uniqueness theorem
View other applications of uniqueness theorem OR Read a survey article on applying uniqueness theorem

Statement

Sheaf-theoretic statement

Consider the sheaf of holomorphic functions on a domain $U$ (an open connected subset in $\mathbb {C}$). For any point $z_{0}\in U$, there is a homomorphism from this sheaf to the sheaf of germs of holomorphic functions at $z_{0}$. This map is injective.

Note: We need $U$ to be connected.

Symbolic statement

Let $U$ be a domain (open connected subset) in $\mathbb {C}$, and $z_{0}\in U$ be a point. Then, the germ of a holomorphic function $f:U\to \mathbb {C}$ at $z_{0}$, determines $f$ completely.