Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 14E / a / Solution

The identity theorem states that if two holomorphic functions on a connected open set $\mathcal{D}$ agree on a subset having an accumulation point in $\mathcal{D}$, then they agree throughout $\mathcal{D}$. In particular, agreement on any nonempty open subset is sufficient.