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 $\sqrt{x_0^2+1}$.