Locally analytic not implies globally analytic

Statement

It is possible to have a function $f$ defined on all of $\mathbb {R}$ such that $f$ is a locally analytic function everywhere on $\mathbb {R}$ but $f$ is not a globally analytic function.

Proof

Consider the function:

$f(x):={\frac {1}{x^{2}+1}}$

This is locally analytic everywhere, but not globally analytic. In fact, the radius of convergence of its power series at any point $x_{0}$ is ${\sqrt {x_{0}^{2}+1}}$.