Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 4 / a

Morera's theorem says that a continuous complex-valued function $f$ on an open set $U\subseteq \mathbb{C}$ is holomorphic iffor every triangle $T$ whose closed interior is contained in $U$. Triangle boundaries are traversed with positive orientation. The continuity hypothesis matters: the zero-integral condition is combined with continuity, not used in isolation.

For the usual piecewise smooth complex path $\gamma$, let $\ell (\gamma )$ denote its finite length. The estimation lemma givesBy uniform convergence, the supremum tends to zero, soThe uniform limit is continuous on the path, so its integral exists. This is uniform convergence and contour integration; the path length is fixed while the functions vary.

Let the common domain be an open set $U$. The uniform limit theorem makes the limit $f$ continuous. For every triangle whose closed interior lies in $U$, Cauchy's integral theorem gives ${\int }_{\partial T}{f}_n=0$. Part (ii) passes this identity to the limit, giving ${\int }_{\partial T}f=0$. Morera's theorem now proves that $f$ is holomorphic.

The same reasoning works for locally uniform convergence, since each triangle boundary is compact. No passage of derivatives through a limit has been assumed.