Holomorphic function is determined by its germ

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.