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 ...
, a real
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
of real numbers is said to be uniformly continuous if there is a positive real number
such that function values over any function domain interval of the size
are as close to each other as we want. In other words, for a uniformly continuous real function of real numbers, if we want function value differences to be less than any positive real number
, then there is a positive real number
such that
at any
and
in any function interval of the size
.
The difference between uniform continuity and (ordinary)
continuity is that, in uniform continuity there is a globally applicable
(the size of a function domain interval over which function value differences are less than
) that depends on only
, while in (ordinary) continuity there is a locally applicable
that depends on the both
and
. So uniform continuity is a stronger continuity condition than continuity; a function that is uniformly continuous is continuous but a function that is continuous is not necessarily uniformly continuous. The concepts of uniform continuity and continuity can be expanded to functions defined between
metric spaces
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 settin ...
.
Continuous functions can fail to be uniformly continuous if they are unbounded on a bounded domain, such as
on
, or if their slopes become unbounded on an infinite domain, such as
on the real (number) line. However, any
Lipschitz map
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
between metric spaces is uniformly continuous, in particular any
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
(distance-preserving map).
Although continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of
neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
s of distinct points, so it requires a metric space, or more generally a
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
.
Definition for functions on metric spaces
For a function
with
metric spaces
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 settin ...
and
, the following definitions of uniform continuity and (ordinary) continuity hold.
Definition of uniform continuity
*
is called uniformly continuous if for every
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 real ...
there exists a real number
such that for every
with
, we have
. The set
for each
is a neighbourhood of
and the set
for each
is a neighbourhood of
by
the definition of a neighbourhood in a metric space.
** If
and
are subsets of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, then
and
can be the
standard one-dimensional Euclidean distance, yielding the following definition: for every real number
there exists a real number
such that for every
,
(where
is a
material conditional
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q is ...
statement saying "if
, then
").
* Equivalently,
is said to be uniformly continuous if
. Here
quantifications (
,
,
, and
) are used.
* Alternatively,
is said to be uniformly continuous if there is a function of all positive real numbers
,
representing the maximum positive real number, such that for every
if
then
.
is a
monotonically non-decreasing function.
Definition of (ordinary) continuity
*
is called continuous
if for every
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 real ...
there exists a real number
such that for every
with
, we have
. The set
is a neighbourhood of
. Thus, (ordinary) continuity is a local property of the function at the point
.
* Equivalently, a function
is said to be continuous if
.
* Alternatively, a function
is said to be continuous if there is a function of all positive real numbers
and
,
representing the maximum positive real number, such that at each
if
satisfies
then
. At every
,
is a monotonically non-decreasing function.
Local continuity versus global uniform continuity
In the definitions, the difference between uniform continuity and
continuity is that, in uniform continuity there is a globally applicable
(the size of a neighbourhood in
over which values of the metric for function values in
are less than
) that depends on only
while in continuity there is a locally applicable
that depends on the both
and
. Continuity is a ''local'' property of a function — that is, a function
is continuous, or not, at a particular point
of the function domain
, and this can be determined by looking at only the values of the function in an arbitrarily small neighbourhood of that point. When we speak of a function being continuous on an
interval, we mean that the function is continuous at every point of the interval. In contrast, uniform continuity is a ''global'' property of
, in the sense that the standard definition of uniform continuity refers to every point of
. On the other hand, it is possible to give a definition that is ''local'' in terms of the natural extension
(the characteristics of which at nonstandard points are determined by the global properties of
), although it is not possible to give a local definition of uniform continuity for an arbitrary hyperreal-valued function, see
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
.
A mathematical definition that a function
is continuous on an interval
and a definition that
is uniformly continuous on
are structurally similar as shown in the following.
Continuity of a function
for
metric spaces
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 settin ...
and
at every point ''
'' of an interval
(i.e., continuity of
on the interval
) is expressed by a formula starting with
quantifications
:
,
(metrics
and
are
and
for
for
the set of real numbers ).
For uniform continuity, the order of the first, second, and third
quantifications (
,
, and
) are rotated:
:
.
Thus for continuity on the interval, one takes an arbitrary point
of the interval'','' and then there must exist a distance
,
:
while for uniform continuity, a single
must work uniformly for all points
of the interval,
:
Properties
Every uniformly continuous function is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
, but the converse does not hold. Consider for instance the continuous function
where
is
the set of real numbers. Given a positive real number
, uniform continuity requires the existence of a positive real number
such that for all
with
, we have
. But
:
and as
goes to be a higher and higher value,
needs to be lower and lower to satisfy
for positive real numbers
and the given
. This means that there is no specifiable (no matter how small it is) positive real number
to satisfy the condition for
to be uniformly continuous so
is not uniformly continuous.
Any
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central oper ...
function (over a compact interval) is uniformly continuous. On the other hand, the
Cantor function
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Th ...
is uniformly continuous but not absolutely continuous.
The image of a
totally bounded In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size ...
subset under a uniformly continuous function is totally bounded. However, the image of a bounded subset of an arbitrary metric space under a uniformly continuous function need not be bounded: as a counterexample, consider the identity function from the integers endowed with the
discrete metric
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
to the integers endowed with the usual
Euclidean metric
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore oc ...
.
The
Heine–Cantor theorem
In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if f \colon M \to N is a continuous function between two metric spaces M and N, and M is compact, then f is uniformly continuous. An important speci ...
asserts that ''every continuous function on a
compact set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
is uniformly continuous''. In particular, ''if a function is continuous on a
closed bounded interval of the real line, it is uniformly continuous on that interval''. The
Darboux integrability of continuous functions follows almost immediately from this theorem.
If a real-valued function
is continuous on
and
exists (and is finite), then
is uniformly continuous. In particular, every element of
, the space of continuous functions on
that vanish at infinity, is uniformly continuous. This is a generalization of the Heine-Cantor theorem mentioned above, since
.
Examples and nonexamples
Examples
* Linear functions
are the simplest examples of uniformly continuous functions.
* Any continuous function on the interval
is also uniformly continuous, since
is a compact set.
* Every function which is differentiable and has bounded derivative is uniformly continuous.
* Every
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
map between two metric spaces is uniformly continuous. More generally, every
Hölder continuous Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of number ...
function is uniformly continuous.
* The
absolute value function is uniformly continuous, despite not being differentiable at
. This shows uniformly continuous functions are not always differentiable.
* Despite being nowhere differentiable, the
Weierstrass function
In mathematics, the Weierstrass function is an example of a real-valued function (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiable nowhere. It is an example of a fractal curve ...
is uniformly continuous.
* Every member of a
uniformly equicontinuous set of functions is uniformly continuous.
Nonexamples
* Functions that are unbounded on a bounded domain are not uniformly continuous. The
tangent function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
is continuous on the interval
but is ''not'' uniformly continuous on that interval, as it goes to infinity as
.
* Functions that have slopes that become unbounded on an infinite domain cannot be uniformly continuous. The exponential function
is continuous everywhere on the real line but is not uniformly continuous on the line, since its derivative tends to infinity as
.
Visualization
For a uniformly continuous function, for every positive real number
there is a positive real number
such that two function values
and
have the maximum distance
whenever
and
are within the maximum distance
. Thus at each point
of the graph, if we draw a rectangle with a height slightly less than
and width a slightly less than
around that point, then the graph lies completely within the height of the rectangle, i.e., the graph do not pass through the top or the bottom side of the rectangle. For functions that are not uniformly continuous, this isn't possible; for these functions, the graph might lie inside the height of the rectangle at some point on the graph but there is a point on the graph where the graph lies above or below the rectangle. (the graph penetrates the top or bottom side of the rectangle.)
File:Gleichmäßig stetige Funktion.svg, For uniformly continuous functions, for each positive real number there is a positive real number such that when we draw a rectangle around each point of the graph with a width slightly less than and a height slightly less than , the graph lies completely inside the height of the rectangle.
File:Nicht gleichmäßig stetige Funktion.svg, For functions that are not uniformly continuous, there is a positive real number such that for every positive real number there is a point on the graph so that when we draw a rectangle with a height slightly less than and a width slightly less than around that point, there is a function value directly above or below the rectangle. There might be a graph point where the graph is completely inside the height of the rectangle but this is not true for every point of the graph.
History
The first published definition of uniform continuity was by
Heine in 1870, and in 1872 he published a proof that a continuous function on an open interval need not be uniformly continuous. The proofs are almost verbatim given by
Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
in his lectures on definite integrals in 1854. The definition of uniform continuity appears earlier in the work of Bolzano where he also proved that continuous functions on an open interval do not need to be uniformly continuous. In addition he also states that a continuous function on a closed interval is uniformly continuous, but he does not give a complete proof.
Other characterizations
Non-standard analysis
In
non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
, a real-valued function ''
'' of a real variable is
microcontinuous at a point ''
'' precisely if the difference
is infinitesimal whenever ''
'' is infinitesimal. Thus ''
'' is continuous on a set ''
'' in
precisely if
is microcontinuous at every real point
. Uniform continuity can be expressed as the condition that (the natural extension of)
is microcontinuous not only at real points in
, but at all points in its non-standard counterpart (natural extension)
in
. Note that there exist hyperreal-valued functions which meet this criterion but are not uniformly continuous, as well as uniformly continuous hyperreal-valued functions which do not meet this criterion, however, such functions cannot be expressed in the form
for any real-valued function
. (see
non-standard calculus
In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered m ...
for more details and examples).
Cauchy continuity
For a function between metric spaces, uniform continuity implies
Cauchy continuity . More specifically, let
be a subset of
. If a function
is uniformly continuous then for every pair of sequences
and
such that
:
we have
:
Relations with the extension problem
Let
be a metric space,
a subset of
'',''
a complete metric space, and
a continuous function. A question to answer: ''When can
be extended to a continuous function on all of
?''
If ''
'' is closed in
, the answer is given by the
Tietze extension theorem
In topology, the Tietze extension theorem (also known as the Tietze–Urysohn–Brouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness ...
. So it is necessary and sufficient to extend ''
'' to the closure of
in
: that is, we may assume without loss of generality that ''
'' is dense in
, and this has the further pleasant consequence that if the extension exists, it is unique. A sufficient condition for
to extend to a continuous function
is that it is
Cauchy-continuous, i.e., the image under ''
'' of a Cauchy sequence remains Cauchy. If
is complete (and thus the completion of ''
''), then every continuous function from ''
'' to a metric space ''
'' is Cauchy-continuous. Therefore when ''
'' is complete, ''
'' extends to a continuous function
if and only if ''
'' is Cauchy-continuous.
It is easy to see that every uniformly continuous function is Cauchy-continuous and thus extends to ''
''. The converse does not hold, since the function
is, as seen above, not uniformly continuous, but it is continuous and thus Cauchy continuous. In general, for functions defined on unbounded spaces like ''
'', uniform continuity is a rather strong condition. It is desirable to have a weaker condition from which to deduce extendability.
For example, suppose
is a real number. At the precalculus level, the function
can be given a precise definition only for rational values of ''
'' (assuming the existence of qth roots of positive real numbers, an application of the
Intermediate Value Theorem). One would like to extend
to a function defined on all of
. The identity
:
shows that ''
'' is not uniformly continuous on the set ''
'' of all rational numbers; however for any bounded interval ''
'' the restriction of ''
'' to
is uniformly continuous, hence Cauchy-continuous, hence
extends to a continuous function on ''
''. But since this holds for every ''
'', there is then a unique extension of ''
'' to a continuous function on all of ''
''.
More generally, a continuous function
whose restriction to every bounded subset of ''
'' is uniformly continuous is extendable to ''
'', and the converse holds if ''
'' is
locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
.
A typical application of the extendability of a uniformly continuous function is the proof of the inverse
Fourier transformation
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
formula. We first prove that the formula is true for test functions, there are densely many of them. We then extend the inverse map to the whole space using the fact that linear map is continuous; thus, uniformly continuous.
Generalization to topological vector spaces
In the special case of two
topological vector spaces
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
and
, the notion of uniform continuity of a map
becomes: for any neighborhood
of zero in
, there exists a neighborhood
of zero in
such that
implies
For
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s
, uniform continuity is equivalent to continuity. This fact is frequently used implicitly in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
to extend a linear map off a dense subspace of a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
.
Generalization to uniform spaces
Just as the most natural and general setting for continuity is
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 points ...
s, the most natural and general setting for the study of ''uniform'' continuity are the
uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s. A function
between uniform spaces is called ''uniformly continuous'' if for every
entourage
An entourage () is an informal group or band of people who are closely associated with a (usually) famous, notorious, or otherwise notable individual. The word can also refer to:
Arts and entertainment
* L'entourage, French hip hop / rap collecti ...
''
'' in ''
'' there exists an entourage ''
'' in ''
'' such that for every
in
we have
in
.
In this setting, it is also true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences.
Each
compact Hausdorff space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
possesses exactly one uniform structure compatible with the topology. A consequence is a generalization of the Heine-Cantor theorem: each continuous function from a compact Hausdorff space to a uniform space is uniformly continuous.
See also
*
*
References
Further reading
* Chapter II is a comprehensive reference of uniform spaces.
*
*
*
*
*
*{{citation, last1=Rusnock, first1=P., last2=Kerr-Lawson, first2=A., title=Bolzano and uniform continuity, journal=Historia Mathematica, volume=32, year=2005, pages=303–311, number=3, doi=10.1016/j.hm.2004.11.003, doi-access=free
Theory of continuous functions
Calculus
Mathematical analysis
General topology