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<p>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.</p>

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Moduli of Continuity and Absolute Continuity: Any Relation?

  • 期刊论文

  • 发表日期：

    2025年1月11日

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  • Matteo Muratori,
  • Jacopo Somaglia

摘要

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.