Uniqueness theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
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Statement

For domains in the complex numbers

Suppose $U$ is a nonempty domain (open connected subset) of $\mathbb {C}$. Then, given two maps $f,g:U\to \mathbb {C}$, exactly one of these two possibilities holds:

• $f\equiv g$ on $U$
• The set of points $z$ for which $f(z)=g(z)$ is a discrete closed subset (i.e. it has no limit points)

For Riemann surfaces

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Related facts

• Holomorphic function is determined by its germ: An easy corollary of the uniqueness theorem