Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Abstract

Given an $o$-minimal expansion $\mathcal M$ of a real closed field $R$ which is not polynomially bounded. Let $\mathcal {TP}^\infty$ denote the definable indefinitely Peano differentiable functions. If we further assume that $\mathcal M$ admits $\mathcal {TP}^{\infty}$ cell decomposition, each definable closed subset $A$ of $R^n$ is the zero-set of a $\mathcal {TP}^{\infty}$ function $f:R^n\rightarrow R$. This implies $\mathcal {TP}^\infty$ approximation of definable continuous functions and gluing of $\mathcal {TP}^\infty$ functions defined on closed definable sets.