Locally piecewise affine functions and their order structure

TL;DR

This paper introduces and characterizes locally piecewise affine functions, demonstrating their density properties in continuous functions and providing a deeper understanding of their structure.

Contribution

It generalizes the concept of piecewise affine functions to locally piecewise affine functions and characterizes them via components and regions.

Findings

Locally piecewise affine functions are the supremum of locally finite sequences of piecewise affine functions.

They are uniformly dense in the space of continuous functions on $\

,

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}.