Absolute continuity of real functions

In mathematics, a real-valued function f of a real variable is absolutely continuous on a specified finite or infinite interval 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 sub-intervals [x_k, y_k], k = 1, ..., n satisfies Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

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Every absolutely continuous function is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous. In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x affect small changes in the output f(x) (continuity), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, a function f : D → R defined on a set D of real numbers with real values is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K ≥ 0 such that for all in D. The smallest such K is called the... Partial plot of a function f. ...

The Cantor function is continuous everywhere but not absolutely continuous; as is the function In mathematics, the Cantor function is a function c : [0,1] → [0,1] defined as follows: Express x in base 3. ...

on a finite interval containing the origin, or the function f(x) = x² on an infinite interval.

• If f is absolutely continuous on a finite interval [a,b], then it is of bounded variation on [a,b].
• If f is absolutely continuous on the interval [a,b], then it has the Luzin N property (that is, for any that λ(L) = 0, it holds that λ(f(L)) = 0, where λ stands for the Lebesgue measure).
• If f is absolutely continuous, then f has a derivative almost everywhere.
• If f is continuous, is of bounded variation and has the Luzin N property, then it is absolutely continuous.

In mathematics, given f, a real-valued function on the interval [a, b] on the real line, the total variation of f on that interval is the supremum running over all partitions P = { x1, ..., xn } of the interval [a, b]. In effect, the total variation is the vertical component of... A function f on the interval [a,b] has the Luzin N property (named after the mathematician Nikolai Luzin, also called Luzin property or N property) if for all that , it holds that , where stands for the Lebesgue measure. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ...

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. It is written as "μ << ν". In mathematics, a measure is a function that assigns a number, e. ... In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ...

If μ is a signed or complex measure, it is said that μ is absolutely continuous with respect to ν if its variation |μ| satisfies |μ| << ν; equivalently, if every set A for which ν(A) = 0 is μ-null. In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. ... In mathematics, or more specifically in measure theory, a complex measure is a generalisation of the concept of measure by letting it have complex values. ... In measure theory, a null set is a set that is negligible for the purposes of the measure in question. ...

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, which implies that there exists a ν-measurable function f taking values in [0,∞], denoted by f = dμ/dν, such that for any ν-measurable set A we have In mathematics, the Radon-Nikodym theorem is a result in functional analysis that states that if a measure Q is absolutely continuous with respect to another sigma-finite measure P then there is a measurable function f, taking values in [0,∞], on the underlying space such that for any measurable...

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 In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras on a topological space X: The minimal σ-algebra containing the open sets. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ...

is locally an absolutely continuous real function. In other words, a function is locally absolutely continuous if and only if its distributional derivative is a measure that is absolutely continuous with respect to the Lebesgue measure.

See also

Singular measure In mathematics, more specifically in measure theory, two measures are mutually singular if their supports are disjoint sets. ...