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# oscillation of a function

###### Definition 1.

Let $f:X\subset\mathbb{R}\to\mathbb{R}$. The oscillation of the function $f$ on the set $X$ is said to be

$\omega(f,X)=\sup_{{a,b\,\in\,X}}|f(b)-f(a)|,$ |

where $a,b$ are arbitrary points in $X$.

# 0.1 Examples

# References

- 1 V., Zorich, Mathematical Analysis I, pp. 131, First Ed., Springer-Verlag, 2004.

Related:

TotalVariation

Type of Math Object:

Definition

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Reference

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26A06*no label found*

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## Corrections

some additions? by pahio ✓

## Comments

## some additions?

Hi Jussi,

>It could be better to write X \subseteq R, since often one thinks that \subset means a proper inclusion.

Well yes, it's a proper inclusion because X could be very small as it is usually included in a sequence of subsets, for instance, a filter basis. In fact this concept is useful to define limit of a function.

> Some mentioning of applications of oscillation?

>What Cauchy's criterion do you mean?

I did put an entry on this theorem. I think this is an interesting approach so I decided to include it in PM.

Cheers my dear friend,

Pedro

## Re: some additions?

Ok, Perucho. I added a link to an elder entry which I think is near to yours.

Cheers,

Jussi

## Re: some additions?

Good! Thank you pal, I give you the Google books reference:

http://www.amazon.com/gp/reader/3540403868/ref=sib_dp_srch_pop?v=search-...

Enter there, It's very interesting.

perucho