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Absolute Continuity of a Function versus Its Modulus of Continuity

The question of whether absolute continuity of a function's modulus of continuity forces the function itself to be absolutely continuous. For Lipschitz functions the two are equivalent, but a monotone continuous counterexample built from the Cantor function shows the implication fails in general: a function can fail to be absolutely continuous while its modulus of continuity is absolutely continuous.

Demerzel Text

2501.00441

For a continuous function f on a compact interval, its minimal modulus of continuity omega_f(delta) records the largest change in f over any pair of points at distance at most delta, and is itself a …