Identity theorem for Riemann surfaces

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

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