# 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

• $\omega(x^{2},\,[-1,2])=4$

• $\omega(x,\,[-1,2])=3$

• $\omega(x,\,(-1,2))=3$

• $\omega(\operatorname{sgn}x\,[-1,2])=2$

• $\omega(\operatorname{sgn}x\,[0,2])=1$

• $\omega(\operatorname{sgn}x\,(0,2])=0$

Cauchy’s criterion can be expressed in terms of this concept.[1]

## References

• 1 V., Zorich, , pp. 131, First Ed., Springer-Verlag, 2004.
Title oscillation of a function OscillationOfAFunction 2013-03-22 17:45:50 2013-03-22 17:45:50 perucho (2192) perucho (2192) 5 perucho (2192) Definition msc 26A06 TotalVariation