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

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 $E$.