Absolute continuity

Common card · No. 204970

In calculus and real analysis, absolute continuity is a regularity property of functions that is stronger than uniform continuity, and hence continuity. The notion of absolute continuity allows one to generalize the fundamental theorem of calculus, originally in the framework of Riemann integration, to Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure. One has the following chains of inclusions for functions over an interval: Lipschitz continuous ⊆ absolutely continuous ⊆ uniformly continuous ⊆ continuous and, for a bounded interval, absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere. Some counterexamples which are continuous but not absolutely continuous:...

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