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).
Definitions
Oscillation of a sequence 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})} :
ω ( a n ) = lim sup n → ∞ a n − lim inf n → ∞ a n {\displaystyle \omega (a_{n})=\limsup _{n\to \infty }a_{n}-\liminf _{n\to \infty }a_{n}} . The oscillation is zero if and only if the sequence converges. It is undefined if lim sup n → ∞ {\displaystyle \limsup _{n\to \infty }} and lim inf n → ∞ {\displaystyle \liminf _{n\to \infty }} are both equal to +∞ or both equal to −∞, that is, if the sequence tends to +∞ or −∞.
Oscillation of a function on an open set 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} :
ω f ( I ) = sup x ∈ I f ( x ) − inf x ∈ I f ( x ) . {\displaystyle \omega _{f}(I)=\sup _{x\in I}f(x)-\inf _{x\in I}f(x).}
More generally, if f : X → R {\displaystyle f:X\to \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
ω f ( U ) = sup x ∈ U f ( x ) − inf x ∈ U f ( x ) . {\displaystyle \omega _{f}(U)=\sup _{x\in U}f(x)-\inf _{x\in U}f(x).}
Oscillation of a function at a point 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 \to 0} of the oscillation of f {\displaystyle f} on an ϵ {\displaystyle \epsilon } -neighborhood of x 0 {\displaystyle x_{0}} :
ω f ( x 0 ) = lim ϵ → 0 ω f ( x 0 − ϵ , x 0 + ϵ ) . {\displaystyle \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 {\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\to \mathbb {R} } is a real-valued function on a metric space, then the oscillation is
… excerpt ends here. Continue reading the full article.
