Let $f:U\to \mathbb{C}$ be any function with $U\subseteq \mathbb{C}$ open. Then $f$ is said to be holomorphic if it is 1;;complex differentiable at all points of $U$.

Later:
Notice how this is similar to the fact that if $(X,{\mathcal{O}}_X)$ is an affine 8. Definition 4.5 Affine Variety, then if two sections agree on some open subset, then they agree on the larger set.