Real analytic functions are defined as functions on the Euclidean spaces with convergent power series at each point.
My question is that, is there some kind of identity theorem for real analytic functions? My book (John Lee's smooth manifolds) says on p.46 that a real-analytic function defined on a connected domain and vanishes on an open set is identically zero.
But I have the impression that this kind of fact holds for holomorphic functions only. Am I missing something?