Oscillation (mathematics)
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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
ω f ( x 0 ) = lim ϵ → 0 ω f ( B ϵ ( x 0 ) ) . {\displaystyle \omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(B_{\epsilon }(x_{0})).}
Examples
- 1 x {\displaystyle {\frac {1}{x}}} has oscillation ∞ at x {\displaystyle x} = 0, and oscillation 0 at other finite x {\displaystyle x} and at −∞ and +∞.
- sin 1 x {\displaystyle \sin {\frac {1}{x}}} (the topologist's sine curve) has oscillation 2 at x {\displaystyle x} = 0, and 0 elsewhere.
- sin x {\displaystyle \sin x} has oscillation 0 at every finite x {\displaystyle x}, and 2 at −∞ and +∞.
- ( − 1 ) x {\displaystyle (-1)^{x}}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. 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
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; in symbols, ω f ( x 0 ) = 0. {\displaystyle \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.
The oscillation is equivalent 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
More generally, if f : X → Y is a function from a topological space X into a metric space Y, then the oscillation of f is defined at each x ∈ X by
ω ( x ) = inf { d i a m ( f ( U ) ) ∣ U i s a n e i g h b o r h o o d o f x } {\displaystyle \omega (x)=\inf \left\{\mathrm {diam} (f(U))\mid U\mathrm {\ is\ a\ neighborhood\ of\ } x\right\}}