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Encyclopedia > Absolutely continuous

## Absolute continuity of real functions

In mathematics, a real_valued function f of a real variable is absolutely continuous if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint intervals [xk, yk], k = 1, ..., n satisfies

then

Every absolutely continuous function is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous.

The Cantor function is continuous everywhere but not absolutely continuous.

## Absolute continuity of measures

If μ and ν are measures on the same measure space (or, more precisely, on the same sigma-algebra) then μ is absolutely continuous with respect to ν if μ(A) = 0 for every set A for which ν(A) = 0. One writes "μ << ν".

The Radon-Nikodym theorem states that if μ is absolutely continuous with respect to ν, and ν is σ-finite, then μ has a density, or "Radon-Nikodym derivative", with respect to ν, i.e., a measurable function f taking values in [0,∞], denoted by f = dμ/dν, such that for any measurable set A we have

## The connection between absolute continuity of real functions and absolute continuity of measures

A measure μ on Borel subsets of the real line is absolutely continuous with respect to Lebesgue measure if and only if the point function

is an absolutely continuous real function.

Results from FactBites:

 The Greek Word Is: 'Kosmos' (629 words) It cannot be possible as possibility is ontologically inferior to the Absolute, yet in the sense of posing a possibility of the absolute we make the absolute seem as though it depends on a contingency of a possibility. The absolute in its singularity condition as the possibility is all- inclusive. Being is that which is not while in the absolute of not being that which is. It is and is not in absolute, absurd singular form.
 Springer Online Reference Works (595 words) Absolute continuity of an integral as a property of the (Lebesgue) integral. Absolute continuity of a measure as a concept in the theory of measures. Absolute continuity of a function is a stronger notion than continuity.
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