# Limited variation

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. ${\ displaystyle \ Omega}$${\ displaystyle BV (\ Omega)}$

## Real functions

### definition

The total variation of a real-valued function defined on a closed interval is the supremum${\ displaystyle f \ colon [a, b] \ to \ mathbb {R}}$

${\ displaystyle \ sup _ {P} \ sum _ {i} | f (x_ {i + 1}) - f (x_ {i}) |,}$

where this supremum is formed over all possible partitions of the interval . The one given here depends on. ${\ displaystyle P = \ {x_ {1}, \ dotsc, x_ {n} \ mid x_ {1} <\ dotsb ${\ displaystyle [a, b]}$${\ displaystyle n}$${\ displaystyle P}$

Riemann-Stieltjes integrates precisely the continuous functions of limited variation . Therefore, a semi-standard can be used: ${\ displaystyle BV [a, b]}$

${\ displaystyle n (f) = \ sup _ {\ varphi} \ int f \, {\ frac {\ mathrm {d} \ varphi} {\ mathrm {d} x}}}$.

This supremum is formed over all functions with a compact carrier and function values ​​in the interval . ${\ displaystyle \ mathrm {\ varphi}}$ ${\ displaystyle [-1.1]}$

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

### example

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. ${\ displaystyle \ textstyle y = \ sin ({\ frac {1} {x}})}$${\ displaystyle x = 0}$ ${\ displaystyle {\ tfrac {1} {x}}}$${\ displaystyle x \ to 0}$

The function

${\ displaystyle f (x) = {\ begin {cases} 0, & {\ text {if}} x = 0 \\ x \ sin (1 / x), & {\ text {if}} x \ neq 0 \ end {cases}}}$

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

${\ displaystyle g (x) = {\ begin {cases} 0, & {\ text {if}} x = 0 \\ x ^ {2} \ sin (1 / x), & {\ text {if}} x \ neq 0 \ end {cases}}}$.

Here the variation of the sine term, which increases for strongly, is dampened enough by the additional power. ${\ displaystyle x \ to 0}$

### 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: ${\ displaystyle BV}$

### 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 ${\ displaystyle \ Omega}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle u \ in L ^ {1} (\ Omega)}$${\ displaystyle BV (\ Omega)}$${\ displaystyle you \ in {\ mathcal {M}} (\ Omega, \ mathbb {R} ^ {n})}$

${\ displaystyle \ int _ {\ Omega} u (x) \, \ operatorname {div} {\ varphi} (x) \ mathrm {d} x = - \ int _ {\ Omega} \ langle \ varphi, you ( x) \ rangle \ qquad {\ text {for all}} {\ varphi} \ in C_ {c} ^ {1} (\ Omega, \ mathbb {R} ^ {n})}$

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. ${\ displaystyle f \ colon [a, b] \ to \ mathbb {R}}$ ${\ displaystyle \ mathbb {R}}$${\ displaystyle f}$${\ displaystyle f}$

## 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 . ${\ displaystyle \ mathbb {R}}$