Oscillation (mathematics)

Amount of variation between extrema

Oscillation of a sequence (shown in blue) is the difference between the limit superior and limit inferior of the sequence.

In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point. As is the case with limits, there are several definitions that put the intuitive concept into a form suitable for a mathematical treatment: oscillation of a sequence of real numbers, oscillation of a real-valued function at a point, and oscillation of a function on an interval (or open set).

## Definitions

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### Oscillation of a sequence

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Let  ( a n ) {\displaystyle (a_{n})} be a sequence of real numbers. The oscillation  ω ( a n ) {\displaystyle \omega (a_{n})} of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of  ( a n ) {\displaystyle (a_{n})} :

ω ( a n ) = lim sup n -> ∞ a n − lim inf n -> ∞ a n {\displaystyle \omega (a_{n})=\limsup _{n\to \infty }a_{n}-\liminf _{n\to \infty }a_{n}} .

The oscillation is zero if and only if the sequence converges. It is undefined if  lim sup n -> ∞ {\displaystyle \limsup _{n\to \infty }} and  lim inf n -> ∞ {\displaystyle \liminf _{n\to \infty }} are both equal to +∞ or both equal to −∞, that is, if the sequence tends to +∞ or −∞.

### Oscillation of a function on an open set

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Let  f {\displaystyle f} be a real-valued function of a real variable. The oscillation of  f {\displaystyle f} on an interval  I {\displaystyle I} in its domain is the difference between the supremum and infimum of  f {\displaystyle f} :

ω f ( I ) = sup x ∈ I f ( x ) − inf x ∈ I f ( x ) . {\displaystyle \omega _{f}(I)=\sup _{x\in I}f(x)-\inf _{x\in I}f(x).}

More generally, if  f : X -> R {\displaystyle f:X\to \mathbb {R} } is a function on a topological space X {\displaystyle X} (such as a metric space), then the oscillation of  f {\displaystyle f} on an open set U {\displaystyle U} is

ω f ( U ) = sup x ∈ U f ( x ) − inf x ∈ U f ( x ) . {\displaystyle \omega _{f}(U)=\sup _{x\in U}f(x)-\inf _{x\in U}f(x).}

### Oscillation of a function at a point

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The oscillation of a function  f {\displaystyle f} of a real variable at a point  x 0 {\displaystyle x_{0}} is defined as the limit as  ϵ -> 0 {\displaystyle \epsilon \to 0} of the oscillation of  f {\displaystyle f} on an  ϵ {\displaystyle \epsilon } -neighborhood of  x 0 {\displaystyle x_{0}} :

ω f ( x 0 ) = lim ϵ -> 0 ω f ( x 0 − ϵ , x 0 + ϵ ) . {\displaystyle \omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(x_{0}-\epsilon ,x_{0}+\epsilon ).}

This is the same as the difference between the limit superior and limit inferior of the function at  x 0 {\displaystyle x_{0}} , provided the point  x 0 {\displaystyle x_{0}} is not excluded from the limits.

More generally, if  f : X -> R {\displaystyle f:X\to \mathbb {R} } is a real-valued function on a metric space, then the oscillation is

ω f ( x 0 ) = lim ϵ -> 0 ω f ( B ϵ ( x 0 ) ) . {\displaystyle \omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(B_{\epsilon }(x_{0})).}

## Examples

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sin (1/x) (the topologist's sine curve) has oscillation 2 at x = 0, and 0 elsewhere.

* 1 x {\displaystyle {\frac {1}{x}}} has oscillation ∞ at  x {\displaystyle x} = 0, and oscillation 0 at other finite  x {\displaystyle x} and at −∞ and +∞.
* sin ⁡ 1 x {\displaystyle \sin {\frac {1}{x}}} (the topologist's sine curve) has oscillation 2 at  x {\displaystyle x} = 0, and 0 elsewhere.
* sin ⁡ x {\displaystyle \sin x} has oscillation 0 at every finite  x {\displaystyle x} , and 2 at −∞ and +∞.
* ( − 1 ) x {\displaystyle (-1)^{x}} or 1, −1, 1, −1, 1, −1... has oscillation 2.

In the last example the sequence is periodic, and any sequence that is periodic without being constant will have non-zero oscillation. However, non-zero oscillation does not usually indicate periodicity.

Geometrically, the graph of an oscillating function on the real numbers follows some path in the xy-plane, without settling into ever-smaller regions. In well-behaved cases the path might look like a loop coming back on itself, that is, periodic behaviour; in the worst cases quite irregular movement covering a whole region.

## Continuity

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Oscillation can be used to define continuity of a function, and is easily equivalent to the usual ε-δ definition (in the case of functions defined everywhere on the real line): a function ƒ is continuous at a point x0 if and only if the oscillation is zero;[1] in symbols,  ω f ( x 0 ) = 0. {\displaystyle \omega _{f}(x_{0})=0.} A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.

For example, in the classification of discontinuities:

* in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
* in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits from the two sides);
* in an essential discontinuity, oscillation measures the failure of a limit to exist.

This definition is useful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε (hence a Gδ set) – and gives a very quick proof of one direction of the Lebesgue integrability condition.[2]

The oscillation is equivalent to the ε-δ definition by a simple re-arrangement, and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given ε0 there is no δ that satisfies the ε-δ definition, then the oscillation is at least ε0, and conversely if for every ε there is a desired δ, the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.

## Generalizations

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More generally, if f : X → Y is a function from a topological space X into a metric space Y, then the oscillation of f is defined at each x ∈ X by

ω ( x ) = inf { d i a m ( f ( U ) ) ∣ U i s a n e i g h b o r h o o d o f x } {\displaystyle \omega (x)=\inf \left\\{\mathrm {diam} (f(U))\mid U\mathrm {\ is\ a\ neighborhood\ of\ } x\right\\}}

## See also

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* Wave equation
* Wave envelope
* Grandi's series
* Bounded mean oscillation

## References

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1. ^ Introduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172
2. ^ Introduction to Real Analysis, updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177

## Further reading

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* Hewitt and Stromberg (1965). Real and abstract analysis. Springer-Verlag. p. 78. ISBN 9780387901381.
* Oxtoby, J (1996). Measure and category (4th ed.). Springer-Verlag. pp. 31–35\. ISBN 978-0-387-90508-2.
* Pugh, C. C. (2002). Real mathematical analysis. New York: Springer. pp. 164–165. ISBN 0-387-95297-7.`{{cite book}}`: CS1 maint: publisher location (link)