Moduli of continuity and absolute continuity: any relation?

TL;DR

The paper investigates whether absolute continuity of a function on a compact interval is equivalent to the absolute continuity of its modulus of continuity $ω_f$, showing the implication fails even for monotone functions. It provides an explicit construction of a monotone continuous $f:[0,7]→[0,7]$ that is not absolutely continuous while $ω_f$ is absolutely continuous, built from a Cantor-function block and a regularizing power function with $α=rac{ ext{log}(2)}{ ext{log}(3)}$. By introducing a related function $g$ that shares the same modulus $ω_f$ but is absolutely continuous, the authors prove that the modulus can be absolutely continuous independently of the original function’s AC status; $ω_g$ is analyzed and shown to be AC through a detailed partitioned-interval argument. The work also presents a simple nonmonotone example with AC modulus and poses an open problem on whether $ω_f$ AC follows from $f$ being absolutely continuous, emphasizing a contrast with the Lipschitz setting and highlighting implications for regularity transfer between $f$ and $ω_f$.

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.

Figures (3)

• Figure 1: Approximate graphs of the functions $f$ (left) and $g$ (right).
• Figure 2: Approximate graph of $\omega_g = \omega_f$.
• Figure 3: Approximate graph of the function $h$.

Theorems & Definitions (11)

• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• proof : Proof of Theorem \ref{['t: mainthm']}