Identity theorem

In real analysis and complex analysis, branches of mathematics, the identity theorem for analytic functions states: given functions f and g analytic on a domain D (open and connected subset of ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$), if f = g on some ${\displaystyle S\subseteq D}$, where ${\displaystyle S}$ has an accumulation point in D, then f = g on D.^[1]

Thus an analytic function is completely determined by its values on a single open neighborhood in D, or even a countable subset of D (provided this contains a converging sequence together with its limit). This is not true in general for real-differentiable functions, even infinitely real-differentiable functions. In comparison, analytic functions are a much more rigid notion. Informally, one sometimes summarizes the theorem by saying analytic functions are "hard" (as opposed to, say, continuous functions which are "soft").

The underpinning fact from which the theorem is established is the expandability of a holomorphic function into its Taylor series.

The connectedness assumption on the domain D is necessary. For example, if D consists of two disjoint open sets, ${\displaystyle f}$ can be ${\displaystyle 0}$ on one open set, and ${\displaystyle 1}$ on another, while ${\displaystyle g}$ is ${\displaystyle 0}$ on one, and ${\displaystyle 2}$ on another.