Radius of convergence of real analytic functions

Let $f: \mathbb{R} \to \mathbb{R}$ be a function that is analytic at every point in $\mathbb{R}$ except at a single point $x_0$.

Radius of Convergence Near $x_0$:

Can we prove that the radius of convergence of the Taylor series of $f$ around any point $x \in \mathbb{R}$ is equal to $|x - x_0|$? In other words, is the radius of convergence of the Taylor series centered at $x$ precisely the distance from $x$ to the point $x_0$ where $f$ is not analytic?

Radius of Convergence on $\mathbb{R}$:

In particular, if $f: \mathbb{R} \to \mathbb{R}$ is analytic at every point in $\mathbb{R}$, does it follow that the Taylor series of $f$ around any point $x_0 \in \mathbb{R}$ converges not only in a neighborhood of $x_0$ but on the entire real line $\mathbb{R}$? More precisely, does this imply that the radius of convergence of the Taylor series centered at $x_0$ is infinite?