In mathematics, the identity theorem for Riemann surfaces is a theorem that states that a holomorphic function is completely determined by its values on any subset of its domain that has a limit point.

Statement of the theorem

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Riemann surfaces, let ${\displaystyle X}$ be connected, and let ${\displaystyle f,g:X\to Y}$ be holomorphic. Suppose that ${\displaystyle f|_{A}=g|_{A))$ for some subset ${\displaystyle A\subseteq X}$ that has a limit point, where ${\displaystyle f|_{A}:A\to Y}$ denotes the restriction of ${\displaystyle f}$ to ${\displaystyle A}$. Then ${\displaystyle f=g}$ (on the whole of ${\displaystyle X}$).