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

In mathematics, oscillation is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or -∞; that is, oscillation is the failure to have a limit, and is also a quantitative measure for that.

Oscillation is defined as the difference (possibly ∞) between the limit superior and limit inferior. It is undefined if both are +∞ or both are -∞ (that is, if the sequence or function tends to +∞ or -∞). For a sequence, the oscillation is defined at infinity, it is zero if and only if the sequence converges. For a function, the oscillation is defined at every limit point in [-∞, +∞] of the domain of the function (apart from the mentioned restriction). It is zero at a point if and only if the function has a finite limit at that point.

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[edit] Examples

As ƒ(x) approaches point P, it oscillates from ƒ(a) to ƒ(b) infinitely many times, and does not converge.
As ƒ(x) approaches point P, it oscillates from ƒ(a) to ƒ(b) infinitely many times, and does not converge.
  • 1/x has oscillation ∞ at x = 0, and oscillation 0 at other finite x and at -∞ and +∞.
  • sin (1/x) 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. On the other hand, non-zero oscillation does not imply 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.

[edit] 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\}

[edit] See also

[edit] References

  • Hewitt and Stromberg (1965). Real and abstract analysis. Springer-Verlag, 78. 
  • Oxtoby, J (1996). Measure and category, 4th ed., Springer-Verlag, 31-35. ISBN 978-0387905082. 
  • Pugh, C. C. (2002). Real mathematical analysis. New York: Springer, pages 164 — 165. ISBN 0387952977.