## Real Analysis Exchange

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

### On Derivatives Vanishing Almost Everywhere on Certain Sets

#### 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