Morera's Theorem

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

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

If, for every simply closed contour $\gamma$ in $D$:


 * $\displaystyle \int_{\gamma} f \left({z}\right) \ \mathrm d z = 0$

then $f$ is analytic on $D$.

Proof
This is the converse of Cauchy's Integral Theorem.