On the level structure of bounded derivatives.

Abstract

We prove: In the space ${\mathcal C}$ of continuous functions on $[0,1]$ under the $sup$ metric, the functions all of whose level sets (in every direction) have measure zero, form a residual subset of ${\mathcal C}$. In the space ${\mathcal D}$ of bounded derivatives of $[0,1]$, the derivatives all of whose level sets are nowhere dense sets of measure zero form a residual subset of ${\mathcal D}$. Moreover, there exists a derivative in ${\mathcal D}$ all of whose level sets have measure zero and one of whose level sets is dense in $[0,1]$.