Mathematics of oscillation

In mathematics, the oscillation of a function or a sequence is a number that quantifies how much a 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). In mathematics, the oscillation of a function or a sequence is a number that quantifies how much a 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 ( a n ) {displaystyle (a_{n})} be a sequence of real numbers. The oscillation ω ( a n ) {displaystyle omega (a_{n})} of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of ( a n ) {displaystyle (a_{n})} : The oscillation is zero if and only if the sequence converges. It is undefined if lim sup n → ∞ {displaystyle limsup _{n o infty }} and lim inf n → ∞ {displaystyle liminf _{n o infty }} are both equal to +∞ or both equal to −∞, that is, if the sequence tends to +∞ or −∞. Let f {displaystyle f} be a real-valued function of a real variable. The oscillation of f {displaystyle f} on an interval I {displaystyle I} in its domain is the difference between the supremum and infimum of f {displaystyle f} : More generally, if f : X → R {displaystyle f:X o mathbb {R} } is a function on a topological space X {displaystyle X} (such as a metric space), then the oscillation of f {displaystyle f} on an open set U {displaystyle U} is The oscillation of a function f {displaystyle f} of a real variable at a point x 0 {displaystyle x_{0}} is defined as the limit as ϵ → 0 {displaystyle epsilon o 0} of the oscillation of f {displaystyle f} on an ϵ {displaystyle epsilon } -neighborhood of x 0 {displaystyle x_{0}} : This is the same as the difference between the limit superior and limit inferior of the function at x 0 {displaystyle x_{0}} , provided the point x 0 {displaystyle x_{0}} is not excluded from the limits. More generally, if f : X → R {displaystyle f:X o mathbb {R} } 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.