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 $\mathbb {R}$.