Morera's Theorem

Theorem
Let $D$ be a simply connected domain in $\C$.

Let $f: D \to \C$ be a continuous function.

Let $f$ be such that:


 * $\ds \int_\gamma \map f z \rd z = 0$

for every simple closed contour $\gamma$ in $D$

Then $f$ is analytic on $D$.

Proof
For a fixed $z_0 \in D$ and $z \in D$ we consider the function:


 * $\ds \map F z = \int_\gamma \map f w \rd w$

where $\gamma$ is any (simple) contour starting at $z_0$ and ending at $z$.

By Primitive of Function on Connected Domain, $F$ is a primitive of $f$.

Since $F$ is analytic and $F' = f$, we conclude that $f$ is analytic as well.

Also see
This is the converse of the Cauchy-Goursat Theorem.