 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.Let formula_1 and formula_2 be Riemann surfaces, let formula_1 be connected, and let formula_4 be holomorphic.Suppose that formula_5 for some subset formula_6 that has a limit point, where formula_7 denotes the restriction of formula_8 to formula_9.