Oscillation (mathematics)

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).

DefinitionsEdit

Oscillation of a sequenceEdit

Let <math>(a_n)</math> be a sequence of real numbers. The oscillation <math>\omega(a_n)</math> of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of <math>(a_n)</math>:

<math>\omega(a_n) = \limsup_{n\to\infty} a_n - \liminf_{n\to\infty} a_n</math>.

The oscillation is zero if and only if the sequence converges. It is undefined if <math>\limsup_{n\to\infty}</math> and <math>\liminf_{n\to\infty}</math> are both equal to +∞ or both equal to −∞, that is, if the sequence tends to +∞ or −∞.

Oscillation of a function on an open setEdit

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

<math>\omega_f(I) = \sup_{x\in I} f(x) - \inf_{x\in I} f(x).</math>

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

<math>\omega_f(U) = \sup_{x\in U} f(x) - \inf_{x\in U}f(x).</math>

Oscillation of a function at a pointEdit

The oscillation of a function <math>f</math> of a real variable at a point <math>x_0</math> is defined as the limit as <math>\epsilon\to 0</math> of the oscillation of <math>f</math> on an <math>\epsilon</math>-neighborhood of <math>x_0</math>:

<math>\omega_f(x_0) = \lim_{\epsilon\to 0} \omega_f(x_0-\epsilon,x_0+\epsilon).</math>

This is the same as the difference between the limit superior and limit inferior of the function at <math>x_0</math>, provided the point <math>x_0</math> is not excluded from the limits.

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

<math>\omega_f(x_0) = \lim_{\epsilon\to 0} \omega_f(B_\epsilon(x_0)).</math>

ExamplesEdit

File:The function sin(1 over x).svg

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

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

ContinuityEdit

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;<ref>Introduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172</ref> in symbols, <math>\omega_f(x_0) = 0.</math> 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.<ref>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</ref>

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.

GeneralizationsEdit

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

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

ReferencesEdit

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