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 $f,g:X\to Y$ 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 $X$).

References

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