Globally analytic function

Definition

Consider a function $f$ defined on all of $\mathbb{R}$. We say that $f$ is globally analytic if it satisfies the following equivalent conditions:

1. There is a point $x_0\in \mathbb{R}$ such that the Taylor series of $f$ at $x_0$ exists and converges to $f$ on all of $\mathbb{R}$.
2. There is a point $x_0\in \mathbb{R}$ such that $f$ has a power series at $x_0$ that converges to $f$ on all of $\mathbb{R}$.
3. For every point $x_0\in \mathbb{R}$, the Taylor series of $f$ at $x_0$ exists and converges to $f$ on all of $\mathbb{R}$.
4. For every point $x_0\in \mathbb{R}$, $f$ has a power series at $x_0$ that converges to $f$ on all of $\mathbb{R}$.