# Oscillation (mathematics)

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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 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).

## Definitions

### Oscillation of a sequence

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

${\displaystyle \omega (a_{n})=\limsup _{n\to \infty }a_{n}-\liminf _{n\to \infty }a_{n}}$.

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

### Oscillation of a function on an open set

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

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

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

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

### Oscillation of a function at a point

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

${\displaystyle \omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(x_{0}-\epsilon ,x_{0}+\epsilon ).}$

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

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

${\displaystyle \omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(B_{\epsilon }(x_{0})).}$

## Examples

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

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, ${\displaystyle \omega _{f}(x_{0})=0.}$ 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 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 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

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

${\displaystyle \omega (x)=\inf \left\{\mathrm {diam} (f(U))\mid U\mathrm {\ is\ a\ neighborhood\ of\ } x\right\}}$

## References

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
• 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.