Luzin-type properties and the difference quotient of a real function

TL;DR

This paper explores the properties of the difference quotient set of a real function and compares it to the function's range, analyzing their measure-theoretic characteristics under various conditions.

Contribution

It introduces the difference quotient set concept and studies its measure-theoretic properties in relation to the function's range on subsets of [0,1].

Findings

Difference quotient set and range can have different measure properties.

Under certain conditions, the difference quotient set can be large or small.

The measure-theoretic behavior depends on the regularity of the function and the set.

Abstract

We introduce the notion of the difference quotient set of a real valued function $f$ on a set $E\subset[0,1]$, and compare this set to the range of $f$ on $E$. We discuss the measure theoretic properties of both the range and the difference quotient set of $f$ over $E$ under different assumptions on $f$ and $E$.