Identity theorem for Riemann surfaces

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 ${\displaystyle X}$).

• Forster, Otto (1981), Lectures on Riemann surfaces, Graduate Text in Mathematics, vol. 81, New-York: Springer Verlag, p. 6, ISBN 0-387-90617-7

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