Definition:Analytic Function/Real Numbers

Definition
Let $f$ be a real function which is smooth on the open interval $\left({a .. b}\right)$.

Let $\xi \in \left({a .. b}\right)$.

Let $\left({c .. d}\right) \subseteq \left({a .. b}\right)$ be an open interval such that:


 * $(1): \quad \xi \in \left({c .. d}\right)$
 * $(2): \quad \displaystyle \forall x \in \left({c .. d}\right): f \left({x}\right) = \sum_{n=0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({x}\right)$

Then $f$ is described as being analytic at the point $\xi$.

That is, a function is analytic at a point if it equals its Taylor series expansion in some interval containing that point.