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.
Moduli of Continuity and Absolute Continuity: Any Relation?
Muratori, Matteo;Somaglia, Jacopo
2025-01-01
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.File in questo prodotto:
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Muratori, Somaglia - Moduli of continuity and absolute continuity any relation.pdf
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