Extend the domain of a real analytic function

Question

Let $f:(a,b)\to \mathbb{R}$ be a real analytic function on $(a,b)$. Is there a real analytic function $g:(c,d)\to\mathbb{R}$, with $(c,d)\supset (a,b)$, such that:

$ g(x)=f(x), \ \forall\ x\in (a,b)? $

Answer 1

Not always. Consider for example $$ f(x)=\frac{1}{x(1-x)},\quad x\in(0,1). $$