Mean value properties for symmetrically differentiable functions
type: Item
instance of: scholarly article
title: Mean value properties for symmetrically differentiable functions
zbMATH Open document ID: 0761.26006
author: Michael J. Evans
author: Shusheng Fu
published in: Real Analysis Exchange
publication date: 1992-10-05T00:00:00Z
review text: The authors define some modifications of Darboux property and mean value property, and examine connections between them for both symmetrically and approximately symmetrically differentiable functions. A function \(f:R\to R\) \((R\) --- the real line) is said to be strongly Darboux \((f\in D^ +)\) if \(f+g\) is Darboux for every linear function \(g(x)=cx\) \((c\in R)\). A function \(f\) is said to be weakly Darboux \((f\in D^ -)\) if for every \(a<b\), every \(C\) strictly between \(f(a)\) and \(f(b)\), and every \(\delta>0\) there is a \(c\in(a-\delta,b+\delta)\) such that \(f(c)=C\). It is shown, e.g., that for a symmetrically differentiable function \(f\in D^ +\) (an approximately continuous, approximately symmetrically differentiable function \(f)\) the following statements are equivalent: 1. \(f^ s\in D^ -\;(f^ s_{ap}\in D^ -)\); 2. for each \(a<b\), there exists \(\xi\in(a,b)\), such that \(f(b)-f(a)=f^ s(\xi)\) \((f(b)-f(a)=f^ s_{ap}(\xi)(b-a))\).
reviewed by: Pavel Kostyrko
Mathematics Subject Classification ID: 26A24
zbMATH DE Number: 68383
zbMATH Keywords: Darboux property
zbMATH Keywords: mean value property
zbMATH Keywords: approximately symmetrically differentiable functions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 68383