An affirmative answer to a problem of Cater

TL;DR

This paper resolves a long-standing open problem by constructing an increasing absolutely continuous function on [0,1] with a dense, countable set of points where its derivative is zero.

Contribution

The paper provides the first explicit example of such a function, answering Cater's problem affirmatively for the first time.

Findings

Constructed an increasing absolutely continuous function with dense zero-derivative points.

Confirmed the existence of functions with dense, countable zero-derivative sets.

Advances understanding of the structure of absolutely continuous functions.

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.