Morera's Theorem

Theorem
Morera's theorem provides the converse of Cauchy's integral theorem.

Suppose that $R$ is a region on $\C$, and $f:R \to\C$ is a continous function.

If, for every closed contour $\gamma$ in $R$ :
 * $\int_{\partial R}f(z) dZ=0$

then $f$ is analytic on $R$.