Absolute continuity

Absolute continuity is a regularity property of functions: for every positive tolerance, sufficiently small total interval length guarantees a correspondingly small total change across any finite collection of disjoint intervals.

Absolutely continuous function: A function whose total change across disjoint intervals is uniformly small when their combined length is small. This is the function-level property defined by absolute continuity.
Lipschitz continuity: A function is Lipschitz continuous when one fixed constant bounds every output difference by the corresponding input distance. Lipschitz continuity imposes a stronger, pointwise rate bound than absolute continuity.
Lebesgue measure: Lebesgue measure assigns lengths, areas, and higher-dimensional volumes to broad classes of sets. The definition measures the combined lengths of the intervals where changes are tested.
Radon–Nikodym theorem: A theorem representing an absolutely continuous measure by integration of a density with respect to another measure. It generalizes the derivative-as-density principle underlying absolute continuity.
Bounded variation: A function has bounded variation when the total size of its changes over partitions is finite. Absolute continuity restricts how variation can concentrate on sets of small length.
Uniform continuity: A function is uniformly continuous when small input distances uniformly ensure small output differences. Uniform continuity controls each interval separately, not the summed changes across disjoint intervals.
Null set: A set has measure zero when it can be covered by countably many intervals of arbitrarily small total length. Absolute continuity prevents function change from being concentrated on such negligible sets.
Absolutely continuous measure: A measure is absolutely continuous with respect to another when every set of measure zero under the latter also has measure zero under the former. This measure-theoretic version extends the same control principle beyond functions.
Lebesgue differentiation theorem: A theorem stating that local averages of an integrable function converge almost everywhere to its pointwise value. It helps recover an absolutely continuous function from its derivative almost everywhere.
Cantor function: A continuous, nondecreasing function that rises from zero to one while having derivative zero almost everywhere. It shows that continuity and an almost-everywhere derivative do not imply absolute continuity.

Linked from 7 pages