On the set of points at which an increasing continuous singular function has a nonzero finite derivative

TL;DR

This paper demonstrates that for increasing continuous singular functions, the set of points with a nonzero finite derivative can be as large as any prescribed null $F_{\sigma}$ subset of [0,1], extending previous results on Hausdorff dimension.

Contribution

It proves that the set of points with a nonzero finite derivative can be any prescribed null $F_{\sigma}$ subset, strengthening earlier Hausdorff dimension results.

Findings

Set of points with nonzero finite derivative can be any null $F_{\sigma}$ subset.

Extends previous results on Hausdorff dimension of such sets.

Shows optimality of the set size for increasing continuous singular functions.

Abstract

Sanchez, Viader, Paradis and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each subinterval of $[0,1]$. We prove a stronger (and optimal) result showing that a set $A_f$ as above can contain any prescribed $F_{\sigma}$ null subset of $[0,1]$.