\begin{definition}[Definition:Analytic Function/Real Numbers]
Let $f$ be a real function which is smooth on the open interval $\openint a b$.
Let $\xi \in \openint a b$.
Let $\openint c d \subseteq \openint a b$ be an open interval such that:
:$(1): \quad \xi \in \openint c d$
:$(2): \quad \ds \forall x \in \openint c d: \map f x = \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } x$
Then $f$ is described as being '''analytic''' at the point $\xi$.
That is, a function is '''analytic''' at a point {{iff}} it equals its Taylor series expansion in some interval containing that point.
\end{definition}