Uniqueness theorem

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

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

Suppose $M$ is a Riemann surface. In other words, $M$ is a connected surface with an atlas of coordinate charts having conformal transition maps. Then, if $f,g:M\to \mathbb{C}$ are holomorphic functions, we either have $f\equiv g$, or the set of points where $f=g$, is a discrete closed subset.

Related facts

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