## Real Analysis Exchange

### On Derivatives Vanishing Almost Everywhere on Certain Sets

F. S. Cater

#### 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