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?
Note that the notion of $C^\infty-$differentiability and analyticity are defined separately
your book is totally, right since a function $f$ being analytic mean that at each point $a$ of the domain there exists $r=r(a)>0$ and a sequence $(a_n)_n$ such that $$f(x) =\sum_{n=1}^{\infty} a_n(x-a)^n~~~~~|x-a|<r $$
and this implies the $C^\infty-$differentiability. The converse is not true for real-function but it is only true for holomorphic functions.
Indeed, you may presumably reconsider your definition within the following function
$$f(x)=e^{-\frac{1}{x^2}}~~~~~f(0)=0$$
This function has a pathology that is $C^\infty(\Bbb R)$ but we have, $$f^{(n)}(0)=0~~~~\forall ~~n\ge0$$
Hence, for all $x\in \Bbb R$$$\sum_{n=1}^{\infty} \frac{f^{(n)}(0)}{n!}x^n =0$$
this function is $C^\infty(\Bbb R)$ non-constant but not analytical near $x=0$.
We can conclude that being real-analytic implies the $C^\infty$ differentiable but the converse is not true.