On the metric Darboux property
type: Item
instance of: scholarly article
title: On the metric Darboux property
zbMATH Open document ID: 0544.26002
DOI: 10.1007/BF01910308
published in: Analysis Mathematica
publication date: 1983-01-01T00:00:00Z
review text: Let \(f\in L_ p(0,1)\) and \[ \omega_ p(f;h):=\sup_{0<t\leq h}\{\int^{1-t}_{0}| f(x+t)-f(x)|^ pdx\}^{1/p} \] be it's modulus of continuity. A function \(f\in S(0,1)\) is named D-function, if for any decomposition an interval \(I\subset(0,1)\) into two measurable sets \(E_ 1\), \(E_ 2\), \(mes E_ i>0,\) one has \( \sup_{E_ 1}f\geq \inf_{E_ 2}f.\) The main result is the following Theorem. If \(\underline{\lim}_{h\to 0}h^{1/p}\omega_ p(f;h)<\infty,\) then \(f=f_ 0+f_ 1,\) where \(f_ 0\) is a D-function, and \(f_ 1\) has left- side and right-side limits at any point of \([0,1].\) The result cannot be improved in terms of the modulus of continuity \(\omega_ p(f;.)\).
Mathematics Subject Classification ID: 26A15
Mathematics Subject Classification ID: 26A16
zbMATH DE Number: 3865618
zbMATH Keywords: metric Darboux property
zbMATH Keywords: \(L_ p\)-space
zbMATH Keywords: modulus of continuity
author: Viktor I. Kolyada
reviewed by: Yuri A. Brudnyi
MaRDI profile type: Publication
cites work: Q3506242
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description: scientific article; zbMATH DE number 3865618