 Solution

Morera's theorem states that a continuous function on an open subset of $\mathbb{C}$ is holomorphic if its integral around every triangular contour, with triangle and interior contained in that open set, is zero.

To prove this, work in any open disk $D$ whose closure lies in $U$, and fix $z_0\in D$. Definewhere the segment lies in $D$ by convexity. The zero integral on the triangle with vertices $z_0,z,z+h$ givesContinuity at $z$ implies that the last integral tends to $f(z)$ as $h\to 0$: its difference from $f(z)$ is bounded by ${\sup }_{0\le t\le 1}|f(z+th)-f(z)|$. Therefore $F^{\prime }=f$ on $D$. The function $F$ is holomorphic, and the allowed theorem that a holomorphic derivative is holomorphic implies that $f=F^{\prime }$ is holomorphic on $D$.

Every point of $U$ lies in such a disk, proving the result throughout $U$:The proof is local and does not require $U$ to be simply connected.