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oscillation of a function
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(Definition)
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Definition 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 $X$
- $\omega(x^2, \, [-1,2])=4$
- $\omega(x, \, [-1,2])=3$
- $\omega(x, \, (-1,2))=3$
- $\omega(\sgn x \, [-1,2])=2$
- $\omega(\sgn x \, [0,2])=1$
- $\omega(\sgn x \, (0,2])=0$
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.
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"oscillation of a function" is owned by perucho. [ full author list (2) ]
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Cross-references: terms, points, function
This is version 2 of oscillation of a function, born on 2008-01-27, modified 2008-04-09.
Object id is 10217, canonical name is OscillationOfAFunction.
Accessed 1661 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) |
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Pending Errata and Addenda
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