Limited variation

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Examples of functions of unlimited variation
Examples of functions of limited variation

In analysis , a function is of limited variation ( limited fluctuation ) if its total variation ( total fluctuation ) is finite, that is, if it does not oscillate indefinitely in a certain way. These terms are closely related to the continuity and the integrability of functions.

The space of all functions of limited variation in the field is denoted by.

Real functions


The total variation of a real-valued function defined on a closed interval is the supremum

where this supremum is formed over all possible partitions of the interval . The one given here depends on.

Riemann-Stieltjes integrates precisely the continuous functions of limited variation . Therefore, a semi-standard can be used:


This supremum is formed over all functions with a compact carrier and function values ​​in the interval .

The semi-norm is consistent with the supremum that defines the restricted variation.


Example of unlimited variation

A simple example of an unbounded variation is the function near . It is clear that the value of the quotient for will grow faster and faster towards ∞ as it approaches 0, and that the sine of this value will run through an infinite number of oscillations. This is shown in the picture on the right.

The function

is also not of limited fluctuation in the interval [0, 1], in contrast to the function:


Here the variation of the sine term, which increases for strongly, is dampened enough by the additional power.


This definition can also be used for complex-valued functions .

BV functions in several variables

Functions of limited variation, or functions, are functions whose distributional derivatives are finite vector-valued Radon measures . More accurate:


Let be an open subset of . A function is of limited variation or element of if its distributional derivative is a finite, signed, vector-valued Radon measure. That is, it exists such that


Connection with rectifiable ways

A continuous function can also be understood as a path in metric space . It holds that is of limited variation if and only if there is a rectifiable path , i.e. if it has a finite length.

Connection with the theory of measure

In the measure theory , the real- / complex-valued functions of bounded variation are exactly the distribution functions of signed / complex Borel dimensions on .