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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the oscillation of a function or a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
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 number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, oscillation of a real-valued function at a point, and oscillation of a function on an interval (or
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
).


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_ a_n - \liminf_ a_n. The oscillation is zero
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
the sequence converges. It is undefined if \limsup_ and \liminf_ 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 In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function (mathematics), function whose domain of a function, domain is the real numbers \mathbb, or ...
. 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_ f(x) - \inf_ f(x). More generally, if f:X\to\mathbb is a function on a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
X (such as a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
), then the oscillation of f on an
open set In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two ...
U is :\omega_f(U) = \sup_ f(x) - \inf_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_ \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 is a real-valued function on a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, then the oscillation is :\omega_f(x_0) = \lim_ \omega_f(B_\epsilon(x_0)).


Examples

*\frac has oscillation ∞ at x = 0, and oscillation 0 at other finite x and at −∞ and +∞. *\sin \frac (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)^xor 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 In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved or n ...
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 ''x''0 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 In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has a ...
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.
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
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 ''ε''0 there is no ''δ'' that satisfies the ''ε''-''δ'' definition, then the oscillation is at least ''ε''0, 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 In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'' into a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
''Y'', then the oscillation of ''f'' is defined at each ''x'' ∈ ''X'' by :\omega(x) = \inf\left\


See also

* Wave equation * Wave envelope * Grandi's series * Bounded mean oscillation


References


Further reading

* * * {{refend Real analysis Limits (mathematics) Sequences and series Functions and mappings Oscillation