oscillation of a function

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

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$\omega(x^{2},\,[-1,2])=4$
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$\omega(x,\,[-1,2])=3$
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$\omega(x,\,(-1,2))=3$
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$\omega(\operatorname{sgn}x\,[-1,2])=2$
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$\omega(\operatorname{sgn}x\,[0,2])=1$
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$\omega(\operatorname{sgn}x\,(0,2])=0$

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

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