Past exam of the geometry course of the University of Oxford / 2007 / B / 8 / iv / Solution

Compactness makes the continuous function $|f|$ attain a maximum. Apply the preceding argument in a chart at a maximizing point to make $f$ constant on a neighborhood. The permitted identity theorem and connectedness then make it constant on all of $S$. Hence every holomorphic function from a compact connected Riemann surface to $\mathbb{C}$ is constant.