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 R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } ), if f = g on some S ⊆ D {\displaystyle S\subseteq D} , where S {\displaystyle S} has an accumulation point in D, then f = g on D. Thus an analytic function is completely determined by its values on a single open neighborhood in D, or even a countable subset of D with an accumulation point (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. 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, f {\displaystyle f} can be 0 {\displaystyle 0} on one open set, and 1 {\displaystyle 1} on another, while g {\displaystyle g} is 0 {\displaystyle 0} on one, and 2 {\displaystyle 2} on another.