Real Analysis Exchange

On Derivatives Vanishing Almost Everywhere on Certain Sets

F. S. Cater

Full-text: Open access


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

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

First available in Project Euclid: 14 May 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


Cater, F. S. On Derivatives Vanishing Almost Everywhere on Certain Sets. Real Anal. Exchange 23 (1999), no. 2, 641--652.

Export citation