Doubts on analytic functions

I have the following doubts about real analytic functions:

1. If $f$ is analytic globally, can its radius of convergence be finite? In other words, if $f$ is analytic at every point, can the Taylor series expansion converge only in a small neighborhood around each point but not on the entire real line $\mathbb{R}$?

2. Does there exist a smooth function (infinitely differentiable for all $x \in \mathbb{R}$) that is not analytic at any point in $\mathbb{R}$?