Assume that $g: [0, \infty) \rightarrow \mathbb{R}$ is $\mathbb{R}$-analytic on $[0, \infty)$ (i.e. for every point of $a \in [0,\infty)$ there exist $R_a>0$ and a real sequence $(a_n)_{n=0}^\infty$ (depending on $a$) such that the series $\sum_{n=0}^\infty a_n (x-a)^n$ is convergent for all $x \in(a-R_a, a+R_a)$ and $\sum_{n=0}^\infty a_n (x-a)^n=f(x)$ for all $x \in(a-R_a, a+R_a)$. Does $g$ can be extended to analytic function on the whole $\mathbb{R}$ ?
P.S. Please without complex analysis, if possible.
Thanks.