In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 called ...
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
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
s, 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, oscillation of a
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real f ...
at a point, and oscillation of a function on an
interval (or
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
).
Definitions
Oscillation of a sequence
Let
be a sequence of real numbers. The oscillation
of that sequence is defined as the difference (possibly infinite) between the
limit superior and limit inferior
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a ...
of
:
:
.
The oscillation is zero if and only if the sequence converges. It is undefined if
and
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
be a real-valued function of a real variable. The oscillation of
on an interval
in its domain is the difference between the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest ...
and
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest lo ...
of
:
:
More generally, if
is a function on a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
(such as a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
), then the oscillation of
on an
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
is
:
Oscillation of a function at a point
The oscillation of a function
of a real variable at a point
is defined as the limit as
of the oscillation of
on an
-neighborhood of
:
:
This is the same as the difference between the limit superior and limit inferior of the function at
, ''provided'' the point
is not excluded from the limits.
More generally, if
is a real-valued function on a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
, then the oscillation is
:
Examples
*
has oscillation ∞ at
= 0, and oscillation 0 at other finite
and at −∞ and +∞.
*
(the
topologist's sine curve
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.
It can be defined as the graph of the functi ...
) has oscillation 2 at
= 0, and 0 elsewhere.
*
has oscillation 0 at every finite
, and 2 at −∞ and +∞.
*
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 ''x''
0 if and only if the oscillation is zero; in symbols,
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
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of a ...
:
* 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" 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 applications to ot ...
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
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of G ...
.
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
In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For ...
,
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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
''X'' into a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
''Y'', then the oscillation of ''f'' is defined at each ''x'' ∈ ''X'' by
:
See also
*
Wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and ...
*
Wave envelope
In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine w ...
*
Grandi's series
*
Bounded mean oscillation In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of functions of bounded mean oscillation (BMO), is a functi ...
References
Further reading
*
*
*
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Real analysis
Limits (mathematics)
Sequences and series
Functions and mappings
Oscillation