Piecewise Constant Roughly Convex Functions

This paper investigates some kinds of roughly convex functions, namely functions having one of the following properties: ρ-convexity (in the sense of Klötzler and Hartwig), δ-convexity and midpoint δ-convexity (in the sense of Hu, Klee, and Larman), γ-convexity and midpoint γ-convexity (in the sense of Phu). Some weaker but equivalent conditions for these kinds of roughly convex functions are stated. In particular, piecewise constant functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$f:\mathbb{R} \to \mathbb{R}$$ \end{document} satisfying f(x) = f([x]) are considered, where [x] denotes the integer part of the real number x. These functions appear in numerical calculation, when an original function g is replaced by f(x):=g([x]) because of discretization. In the present paper, we answer the question of when and in what sense such a function f is roughly convex.