Moduli of continuity and absolute continuity: any relation?

TL;DR

This paper constructs a specific function demonstrating that absolute continuity of a function does not necessarily imply the absolute continuity of its modulus of continuity, challenging a common assumption.

Contribution

It provides a counterexample showing the lack of equivalence between a function's absolute continuity and that of its modulus of continuity.

Findings

Constructed a monotone, continuous, not absolutely continuous function

Showed the minimal modulus of continuity can be absolutely continuous

Contrasted with the property of Lipschitz functions

Abstract

We construct a monotone, continuous, but not absolutely continuous function whose minimal modulus of continuity is absolutely continuous. In particular, we establish that there is no equivalence between the absolute continuity of a function and the absolute continuity of its modulus of continuity, in contrast with a well-known property of Lipschitz functions.