Morera's 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

Suppose $U$ is a domain (open connected subset) in $\mathbb {C}$ and $f:U\to \mathbb {C}$ is a continuous function with the property that for every triangle $\triangle \subset U$, we have:

$\int _{\partial \triangle }f(z)dz=0$

Then, $f$ is a holomorphic function.

Related facts

• Goursat's integral lemma is a converse of sorts