
In
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
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 of
:
:
.
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
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
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
on an interval
in its domain is the difference between the
supremum and
infimum of
:
:
More generally, if
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 ...
(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
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 ...
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 (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
:
Examples
*
has oscillation ∞ at
= 0, and oscillation 0 at other finite
and at −∞ and +∞.
*
(the
topologist's sine curve) 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
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,
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
:
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