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).
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 -?.
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
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.
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; 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:
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.
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.