Is Analytic Continuation possible if we assume real analytic function?

Question

I come across proof of analytic continuation.
As follows

In this theorem Author uses only analytic property i.e existence of power series around some point.

I was thinking if in case of real function if we have such function i.e which has power series exapansion then it should posses analytical continution.

IS my right /Or where I am Missing?
ANy Help will be appreciated Any Help will be appreciated

Answer 1

Yes, this is true for analytic functions. Whether we're working over the real numbers or over the complex numbers is irrelevant.

However, this has nothing to do with analytic continuation. The statement that we are talking about here is the identity theorem.