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, 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.

[edit] Examples

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

[edit] See also

• Grandi's series
• Bounded mean oscillation

[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.