Locally analytic not implies globally analytic

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Statement

It is possible to have a function f defined on all of \R such that f is a locally analytic function everywhere on \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}.