Holomorphic functions that agree on a set with an interior limit point agree everywhere on the connected domain.Suggest a slogan
Let \(\Omega\subseteq\CC\) be a connected open set and let \(f,g\) be holomorphic on \(\Omega\). If \(f=g\) on a subset of \(\Omega\) that has a limit point in \(\Omega\), then \(f=g\) on \(\Omega\).
The function \(f-g\) is holomorphic on \(\Omega\), and its zeros have a limit point in \(\Omega\). The zeros of a holomorphic function on a connected open set that is not identically zero are isolated, so \(f-g\equiv0\).