Mathematics of oscillation

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

The mathematics of oscillation deals with the quantification of the amount that a sequence or function tends to move between extremes. There are several related notions: 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[edit]

Oscillation of a sequence[edit]

If (a_n) is a sequence of real numbers, then the oscillation of is defined as the difference (possibly ∞) between the limit superior and limit inferior of a_n:

\omega(a_n) = \lim\sup a_n - \lim\inf a_n.

It is undefined if both are +∞ or both are −∞, that is, if the sequence tends to +∞ or to −∞. The oscillation is zero if and only if the sequence converges.

Oscillation of a function on an open set[edit]

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

\omega_f(I) = \sup_{x\in I} f(x) - \inf_{x\in I} f(x).

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

\omega_f(U) = \sup_{x\in U} f(x) - \inf_{x\in U}f(x).

Oscillation of a function at a point[edit]

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

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

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

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

\omega_f(x_0) = \lim_{\epsilon\to 0} \omega_f(B_\epsilon(x_0)).

Examples[edit]

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[edit]

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, \omega_f(x_0) = 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 Gδ 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[edit]

More generally, if f : XY is a function from a topological space X into a metric space Y, then the oscillation of f is defined at each xX by

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

See also[edit]

References[edit]

  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 978-0-387-90508-2. 
  • Pugh, C. C. (2002). Real mathematical analysis. New York: Springer. pp. 164–165. ISBN 0-387-95297-7.