oscillation of a function

Definition 1.

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

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

where $a\mathrm{,}b$ are arbitrary points in $X$.

0.1 Examples

• $\omega (x^2,[-1,2])=4$

• $\omega (x,[-1,2])=3$

• $\omega (x,(-1,2))=3$

• $\omega (\mathrm{sgn}x[-1,2])=2$

• $\omega (\mathrm{sgn}x[0,2])=1$

• $\omega (\mathrm{sgn}x(0,2])=0$

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