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 \mathbb{R}
\left( |x - x_0| < R \quad \implies \quad f(x) = \sum_{n = 0}^{\infty} a_n (x - x_0)^n \right)
\end{equation}
