Let $\mathbb{R}$ be the D4369: Standard real metric space such that

(i)	$E \subseteq \mathbb{R}$ is a D78: Subset of $\mathbb{R}$
(ii)	\begin{equation} E \neq \emptyset \end{equation}
(iii)	$x_0 \in E$ is an D1387: Interior point of $E$ in $\mathbb{R}$

A D4364: Real function $f : E \to \mathbb{R}$ is analytic at $x_0$ if and only if \begin{equation} \exists \, R > 0, a \in \mathbb{R}^{\mathbb{N}} : \forall \, x \in B(x_0, R) : f(x) = \sum_{n = 0}^{\infty} a_n (x - x_0)^n \end{equation}