Oscillation (mathematics)
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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).

Definitions

Oscillation of a sequence

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

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

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

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 ?-? definition (in the case of functions defined everywhere on the real line): a function ? is continuous at a point x0 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 ? (hence a G? set) - and gives a very quick proof of one direction of the Lebesgue integrability condition.[2]

The oscillation is equivalence to the ?-? 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 ?0 there is no ? that satisfies the ?-? definition, then the oscillation is at least ?0, and conversely if for every ? there is a desired ?, 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

See also

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.

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Oscillation_(mathematics)
 



 



 
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