Let \(f\colon U\subset \mathbb{R}\to \mathbb{R}\) be smooth. The function \(f\) is real analytic function if and only if for every \(x_0\in U\) there is a subset \(V\subset U\) with \(x_0\in V\), and constants \(C>0\) and \(R>0\) such that for all \(x\in V\) and \(k\in \mathbb{N}\)

\begin{equation*} \lvert f^{(k)}(x)\rvert\le C \frac{k!}{R^k}. \end{equation*}

[1, Proposition 1.2.12]

References Link to heading

1. S. Krantz and H. Parks, A Primer of Real Analytic Functions. Boston, MA: Birkhäuser Boston, 2002. doi:10.1007/978-0-8176-8134-0