Absolute continuity

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 obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration. This relationship is commonly characterized (by the fundamental theorem of calculus) in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of 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. We have 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.