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").^{[citation needed]}

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.