Locally piecewise affine functions and their order structure

Abstract

Piecewise affine functions on subsets of ${\mathbb{R}}^m$ were studied in \cite{Ovchinnikov:02,Aliprantis:06a,Aliprantis:07a,Aliprantis:07}. In this paper we study a more general concept of a locally piecewise affine function. We characterize locally piecewise affine functions in terms of components and regions. We prove that a positive function is locally piecewise affine iff it is the supremum of a locally finite sequence of piecewise affine functions. We prove that locally piecewise affine functions are uniformly dense in $C({\mathbb{R}}^m)$, while piecewise affine functions are sequentially order dense in $C({\mathbb{R}}^m)$. This paper is partially based on \cite{Adeeb:14}.

11 pages