Real Analysis Exchange

On Derivatives Vanishing Almost Everywhere on Certain Sets

F. S. Cater

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Abstract

Let \(g\) be a measurable real valued function on a bounded, measurable subset of the real line. We prove that if \(g(E)\) has measure 0, then 0 is one of the derived numbers of \(g\) at almost every point in \(E\). We find a function \(H\) on the real line that is nondecreasing and closely associated with \(G\), such that if \(g(E)\) has measure 0, the \(H'\) vanishes almost everywhere. Moreover, if \(g\) is an \(N\)-function on \(E\) and if \(H'\) vanishes almost everywhere, then \(g(E)\) has measure 0.

Article information

Source
Real Anal. Exchange, Volume 23, Number 2 (1999), 641-652.

Dates
First available in Project Euclid: 14 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.rae/1337001371

Mathematical Reviews number (MathSciNet)
MR1640000

Zentralblatt MATH identifier
0943.26016

Subjects
Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

Keywords
{derivative} {approximate derivative} {derived number} {\(N\)-function} {measure} {bounded variation}

Citation

Cater, F. S. On Derivatives Vanishing Almost Everywhere on Certain Sets. Real Anal. Exchange 23 (1999), no. 2, 641--652. https://projecteuclid.org/euclid.rae/1337001371


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