Regularity of locally Lipschitz functions on the line
type: Item
instance of: scholarly article
title: Regularity of locally Lipschitz functions on the line
zbMATH Open document ID: 0843.26003
author: Scott Sciffer
published in: Real Analysis Exchange
publication date: 1996-08-19T00:00:00Z
review text: We say that \(f\) is regular at \(x\) if \(f^0_+(x)= f_+(x)\) and \(f^0_-(x)= f_-(x)\) (where \(f^0_+\), \(f^0_-\) denote right and left Clarke derivatives of \(f\), respectively, and \(f_+\), \(f_-\) are Dini derivatives. First, the author mentions that a regular locally Lipschitz function on an open interval is differentiable at all but a countable set of points. In the next parts, the author constructs some set \(E_1\) for which \(\int^x_0\chi_{E_1}(t) d\lambda\) (\(\lambda\) denotes the Lebesgue measure) is a locally Lipschitz and nowhere regular function mapping \((0, 1)\) into \(\mathbb{R}\).
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 58C20
Mathematics Subject Classification ID: 49J52
zbMATH DE Number: 801035
zbMATH Keywords: Clarke derivatives
zbMATH Keywords: Dini derivatives
zbMATH Keywords: regular locally Lipschitz function
reviewed by: Ryszard Pawlak
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 801035