Regularity of Lipschitz functions on the line.
type: Item
instance of: scholarly article
title: Regularity of Lipschitz functions on the line.
zbMATH Open document ID: 1057.26008
author: David Preiss
author: Louise Rolland
published in: Real Analysis Exchange
publication date: 2004-08-06T00:00:00Z
review text: The authors note a gap in Sciffer's construction of an everywhere irregular Lipschitz function of the real line and give their own construction. The Dini derivatives are denoted by \(D^+\), \(D_+\), \(D^-\), \(D_-\). The Clarke derivatives are \(S^+f(x)=\limsup_{y\to x+,h\to0}(f(y+h)-f(y))/h\), \(S_+f(x)=\liminf_{y\to x+,h\to0}(f(y+h)-f(y))/h\), \(S^-f(x)=\limsup_{y\to x-,h\to0}(f(y+h)-f(y))/h\), \(S_-f(x)=\liminf_{y\to x-,h\to0}(f(y+h)-f(y))/h\). The Lipschitz function constructed in this paper has \(S^+f(x)>D_+f(x)\), \(D^+f(x)>S_+f(x)\), \(S^-f(x)>D_-f(x)\) and \(D^-f(x)>S_-f(x)\) at each \(x\in\mathbb R\). The example is optimal also quantitatively.
reviewed by: Jan Malý
Mathematics Subject Classification ID: 26A27
Mathematics Subject Classification ID: 26A16
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 2084034
zbMATH Keywords: Dini derivatives
zbMATH Keywords: Clarke derivatives
zbMATH Keywords: nondifferentiable functions
zbMATH Keywords: Lipschitz functions
MaRDI profile type: Publication
full work available at URL: realanalexch.28.1.0227
OpenAlex ID: W1555625181
DOI: 10.14321/REALANALEXCH.28.1.0227
description: scientific article; zbMATH DE number 2084034