Oscillation (mathematics)

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

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 $(a_{n})$ be a sequence of real numbers. The oscillation $\omega (a_{n})$ of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of $(a_{n})$ :

\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 $\limsup _{n\to \infty }$ and $\liminf _{n\to \infty }$ 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 $f$ 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

${\frac {1}{x}}$ has oscillation ∞ at $x$ = 0, and oscillation 0 at other finite $x$ and at −∞ and ∞.
$\sin {\frac {1}{x}}$ (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 ∞.
$(-1)^{x}$ 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;^[1] 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.^[2]

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

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

References

^ Introduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172
^ Introduction to Real Analysis, updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177