Globally analytic function

Definition

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

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