We introduce the notion of R-analytic functions. These are definable in an o-minimal expansion of a real closed field R and are locally the restriction of a K-differentiable function where K=R[-1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=R[\sqrt{-1}]$$\end{document} is the algebraic closure of R. The class of these functions in this general setting exhibits the nice properties of real analytic functions. We also define strongly R-analytic functions. These are globally the restriction of a K-differentiable function. We show that (...) in arbitrary models of important o-minimal theories strongly R-analytic functions abound and that the concept of analytic cell decomposition can be transferred to non-standard models. (shrink)