Oscillation (mathematics)

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 −∞. Let be a real-valued function of a real variable. The oscillation of on an interval in its domain is the difference between the supremum and infimum of : More generally, if is a function on a topological space (such as a metric space), then the oscillation of on an open set is 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 has oscillation ∞ at = 0, and oscillation 0 at other finite and at −∞ and +∞. (the topologist's sine curve) has oscillation 2 at = 0, and 0 elsewhere. has oscillation 0 at every finite , and 2 at −∞ and +∞. or 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.