Past exam of the algebra and analysis II for mathematics and philosophy course of the University of Oxford / 2013 / 4 / b

Cauchy's integral theorem says that if $f$ is holomorphic on an open set $U$ and a closed, piecewise smooth contour $\gamma$ is null-homotopic in $U$, thenIn particular, this holds for every closed contour when $U$ is simply connected, and for a simple closed contour whose boundary and enclosed region lie in $U$. For the later arguments it suffices to use the triangular version: the integral around a triangle is zero if the triangle and its interior lie in $U$ and $f$ is holomorphic there.

Both the holomorphicity condition and a suitable condition excluding holes inside the contour matter. Continuity alone, which is the starting hypothesis for this question, does not imply the theorem's conclusion; nor does holomorphicity on an arbitrary punctured open set imply zero integral around every loop.