An affirmative answer to a problem of Cater

TL;DR

This paper resolves Cater's problem by constructing a strictly increasing absolutely continuous function $f$ on $[0,1]$ whose derivative vanishes on a set that is both countable and dense. The construction uses an auxiliary increasing continuous function $F$ guaranteed by Theorem A (due to V. M. Tzodiks–Tzo), which imposes $F'(x)=+\infty$ on a chosen dense countable set $R$ while controlling derivatives off $R$, and then takes $f$ to be the inverse of a normalized version of $F$. Consequently, the zero set $\{x: f'(x)=0\}$ is exactly the image $F(R)$, which is countable and dense. The paper also points to a sharp contrast: no increasing continuous function can have the lower derivative zero set be countable and dense, a result to be addressed in a subsequent publication. Overall, the work contributes a precise example showing delicate derivative behavior for absolutely continuous functions within differentiability theory.

Abstract

Does there exist an increasing absolutely continuous function, $f: [0,1] \rightarrow \mathbb R$ such that $\{x: f'(x)=0\}$ is both countable and dense? This problem was proposed by F.S. Cater about two decades ago. We give an affirmative answer to the problem.