# 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

### definition

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

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.

### Extensions

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:

### definition

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

applies.

## 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 .

## literature

- Jürgen Elstrodt : Measure and integration theory . 5th edition. Springer, Berlin 2007, ISBN 978-3-540-49977-0 .
- Gerald Teschl : Topics in Real and Functional Analysis . 2011 ( free online version ).
- Luigi Ambrosio , Nicola Fusco and Diego Pallara : Functions of Bounded Variation and Free Discontinuity Problems . Oxford 2000.