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Definition D4715
Analytic real function

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) $$E \neq \emptyset$$ (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 $$\exists \, R > 0, a \in \mathbb{R}^{\mathbb{N}} : \forall \, x \in B(x_0, R) : f(x) = \sum_{n = 0}^{\infty} a_n (x - x_0)^n$$

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) $$E \neq \emptyset$$ (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 $$\exists \, R > 0, a \in \mathbb{R}^{\mathbb{N}} : \forall \, x \in \mathbb{R} \left( |x - x_0| < R \quad \implies \quad f(x) = \sum_{n = 0}^{\infty} a_n (x - x_0)^n \right)$$