Differentiable restrictions of continuous functions
type: Item
instance of: scholarly article
title: Differentiable restrictions of continuous functions
zbMATH Open document ID: 0558.26005
DOI: 10.1007/BF01950290
published in: Acta Mathematica Hungarica
publication date: 1984-01-01T00:00:00Z
review text: Let \(f: P\to R\). If f is continuous, P is perfect and \(m(P)>0,\) then for every \(\epsilon >0\) there is a perfect \(Q\subset P\) such that \(m(P- Q)<\epsilon\) and \(f|_ Q\) is differentiable. If f is measurable, and \(m(P)>0,\) then either (1) there is a perfect \(Q\subset P\) such that \(Q\neq \emptyset\) and \(f|_ Q\) is infinitely differentiable on Q and \((f|_ Q)^{(n)}=0\) for n sufficiently large, or (2) for every \(\epsilon >0\) there is a perfect \(Q\subset P\) such that \(m(P-Q)<\epsilon\) and \(f|_ Q\) is infinitely differentiable on Q.
Mathematics Subject Classification ID: 26A24
Mathematics Subject Classification ID: 26A15
zbMATH DE Number: 3889828
zbMATH Keywords: continuous function
zbMATH Keywords: differentiable restriction
zbMATH Keywords: perfect subset
author: M. Laczkovich
reviewed by: Zbigniew Grande
MaRDI profile type: Publication
cites work: On the Differentiability Structure of Real Functions
cites work: On level sets of a continuous nowhere monotone function
cites work: On The Roots of Equation f(x) = ξ Where f(x) is Real and Continuous in (a, b) but Monotonic in no Subinterval of (a, b)
cites work: Baire 1 functions, approximately continuous functions and derivatives
cites work: Q5768777
description: scientific article; zbMATH DE number 3889828