# Oscillation (mathematics)

In mathematics, the **oscillation** of a function or a sequence is a number that quantifies how much a 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).

## Contents

## Definitions[edit]

### Oscillation of a sequence[edit]

Let be a sequence of real numbers. The oscillation of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of :

- .

The oscillation is zero if and only if the sequence converges. It is undefined if and are both equal to +∞ or both equal to −∞, that is, if the sequence tends to +∞ or −∞.

### Oscillation of a function on an open set[edit]

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

More generally, if is a function on a topological space (such as a metric space), then the oscillation of on an open set is

### Oscillation of a function at a point[edit]

The oscillation of a function of a real variable at a point is defined as the limit as of the oscillation of on an -neighborhood of :

This is the same as the difference between the limit superior and limit inferior of the function at , *provided* the point is not excluded from the limits.

More generally, if is a real-valued function on a metric space, then the oscillation is

## Examples[edit]

- 1/
*x*has oscillation ∞ at*x*= 0, and oscillation 0 at other finite*x*and at −∞ and +∞. - sin (1/
*x*) (the topologist's sine curve) has oscillation 2 at*x*= 0, and 0 elsewhere. - sin
*x*has oscillation 0 at every finite*x*, and 2 at −∞ and +∞. - The sequence 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[edit]

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 *x*_{0} if and only if the oscillation is zero;^{[1]} in symbols, 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 G_{δ} set) – and gives a very quick proof of one direction of the Lebesgue integrability condition.^{[2]}

The oscillation is equivalence 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 *ε*_{0} there is no *δ* that satisfies the *ε*-*δ* definition, then the oscillation is at least *ε*_{0}, 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[edit]

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

## See also[edit]

## References[edit]

**^***Introduction to Real Analysis,*updated April 2010, William F. Trench, Theorem 3.5.2, p. 172**^***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

- Hewitt and Stromberg (1965).
*Real and abstract analysis*. Springer-Verlag. p. 78. - 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.