Morera's Theorem

If $f(z)$ is continuous in a simply connected region $D$ and satisfies

$$\int_\gamma f\,dz = 0$$

for all closed Contours $\gamma$ in $D$, then $f(z)$ is Analytic in $D$.

See also Cauchy Integral Theorem

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 373-374, 1985.