 Solution

Morera's theorem states that a continuous complex-valued function on an open set is holomorphic if its integral around every triangle whose closed interior lies in the set is zero. It is enough to impose this condition locally on disks.

Fix a disk $B$ contained in the open set and a base point $z_0\in B$. Define $F(z)$ as the integral of $f$ along the straight segment from $z_0$ to $z$. The zero triangle integral gives, for sufficiently small $h$,Continuity makes the quotient by $h$ tend to $f(z)$, so $F$ is holomorphic and $F^{\prime }=f$. To justify that its derivative is holomorphic, apply the Cauchy integral formula for $F$ on a smaller circle:The right side may be differentiated again inside the circle because its denominator stays uniformly away from zero on the contour. Thus $F^{\prime }=f$ is holomorphic on $B$. Such disks cover the open set, proving the theorem.