Oscillation (mathematics)

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).

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

Definitions

Oscillation of a sequence

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

\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 $\limsup _{n\to \infty }$ and $\liminf _{n\to \infty }$ 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 $f$ be a real-valued function of a real variable. The oscillation of $f$ on an interval $I$ in its domain is the difference between the supremum and infimum of $f$ :

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

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

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

\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 $x_{0}$ , provided the point $x_{0}$ is not excluded from the limits.

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

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

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

References

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.

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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