Intrinsic metric

Oscillation of a sequence (shown in blue) is the difference between the limit superior and limit inferior of the sequence.

In mathematics, oscillation quantifies the amount that a sequence or function tends to move between extremes. There are several related notions: 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

If ($a_{n}$) is a sequence of real numbers, then the oscillation of is defined as the difference (possibly ∞) between the limit superior and limit inferior of $a_{n}$:

$\omega (a_{n})=\lim \sup a_{n}-\lim \inf a_{n}.$

It is undefined if both are +∞ or both are −∞, that is, if the sequence tends to +∞ or to −∞. The oscillation is zero if and only if the sequence converges.

Oscillation of a function on an open set

Let $f$ be a real-valued function of a real variable. The oscillation of $f$ on an interval $I$ in its domain is the difference between the supremum and infimum of $f$:

$\omega _{f}(I)=\sup _{x\in I}f(x)-\inf _{x\in I}f(x).$

More generally, if $f:X\to \mathbb {R}$ is a function on a topological space $X$ (such as a metric space), then the oscillation of $f$ on an open set $U$ is

$\omega _{f}(U)=\sup _{x\in U}f(x)-\inf _{x\in U}f(x).$

Oscillation of a function at a point

The oscillation of a function $f$ of a real variable at a point $x_{0}$ is defined as the limit as $\epsilon \to 0$ of the oscillation of $f$ on an $\epsilon$-neighborhood of $x_{0}$:

$\omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(x_{0}-\epsilon ,x_{0}+\epsilon ).$

This is the same as the difference between the limit superior and limit inferior of the function at $x_{0}$, provided the point $x_{0}$ is not excluded from the limits.

More generally, if $f:X\to \mathbb {R}$ is a real-valued function on a metric space, then the oscillation is

$\omega _{f}(x_{0})=\lim _{\epsilon \to 0}\omega _{f}(B_{\epsilon }(x_{0})).$

Examples

As the argument of ƒ approaches point P, ƒ oscillates from ƒ(a) to ƒ(b) infinitely many times, and does not converge.

• 1/x has oscillation ∞ at x = 0, and oscillation 0 at other finite x and at −∞ and +∞.
• sin (1/x) (the topologist's sine curve) has oscillation 2 at x = 0, and 0 elsewhere.
• sin x has oscillation 0 at every finite x, and 2 at −∞ and +∞.
• The sequence 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, $\omega _{f}(x_{0})=0.$ 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 G_δ set) – and gives a very quick proof of one direction of the Lebesgue integrability condition.

The oscillation is equivalence 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

$\omega (x)=\inf \left\{\mathrm {diam} (f(U))\mid U\mathrm {\ is\ a\ neighborhood\ of\ } x\right\}$

See also

• Grandi's series
• Bounded mean oscillation

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534