Editing Absolute continuity (section)

{{Short description|Form of continuity for functions}}
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In [[calculus]] and [[real analysis]], '''absolute continuity''' is a [[smoothness (mathematics)|smoothness]] property of [[function (mathematics)|function]]s that is stronger than [[continuous function|continuity]] and [[uniform continuity]]. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of [[calculus]]—[[derivative|differentiation]] and [[integral|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 a [[compact space|compact]] subset''' of the real line:

: ''absolutely continuous'' ⊆ ''[[uniformly continuous]]'' <math>=</math> ''[[Continuous function|continuous]]''<!--All continuous functions on a compact domain are uniformly continuous-->

and, for a compact interval,

: '''[[continuously differentiable]]''' ⊆ '''[[Lipschitz continuous]]''' ⊆ '''absolutely continuous''' ⊆ '''[[bounded variation]]'''  ⊆ '''[[Differentiable function|differentiable]] [[almost everywhere]]'''.