Oscillation (mathematics)

For other uses, see Oscillation (differential equation).

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 difference between the superior and inferior limits of the sequence or function is in one of the indeterminate forms +∞ + (-∞) 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.

Examples

As the argument of ƒ approaches point P, ƒ 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) (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, ω_f(x₀) = 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

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

See also

• Grandi's series
• Bounded mean oscillation

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. ed.). Springer-Verlag. pp. 31–35. ISBN 978-0387905082. 
• Pugh, C. C. (2002). Real mathematical analysis. New York: Springer. pp. 164–165. ISBN 0387952977.