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 ...
, a continuous function is a
function such that a small variation of the
argument
An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
induces a small variation of the
value of the function. This implies there are no abrupt changes in value, known as ''
discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on
intuitive notions of continuity and considered only continuous functions. The
epsilon–delta definition of a limit was introduced to formalize the definition of continuity.
Continuity is one of the core concepts of
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
and
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
, where arguments and values of functions are
real and
complex numbers. The concept has been generalized to functions
between metric spaces and
between topological spaces. The latter are the most general continuous functions, and their definition is the basis of
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
.
A stronger form of continuity is
uniform continuity. In
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, especially in
domain theory, a related concept of continuity is
Scott continuity.
As an example, the function denoting the height of a growing flower at time would be considered continuous. In contrast, the function denoting the amount of money in a bank account at time would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.
History
A form of the
epsilon–delta definition of continuity was first given by
Bernard Bolzano in 1817.
Augustin-Louis Cauchy defined continuity of
as follows: an infinitely small increment
of the independent variable ''x'' always produces an infinitely small change
of the dependent variable ''y'' (see e.g. ''
Cours d'Analyse'', p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see
microcontinuity). The formal definition and the distinction between pointwise continuity and
uniform continuity were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano,
Karl Weierstrass denied continuity of a function at a point ''c'' unless it was defined at and on both sides of ''c'', but
Édouard Goursat allowed the function to be defined only at and on one side of ''c'', and
Camille Jordan allowed it even if the function was defined only at ''c''. All three of those nonequivalent definitions of pointwise continuity are still in use.
Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by
Peter Gustav Lejeune Dirichlet in 1854.
Real functions
Definition

A
real function that is a
function from
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 to real numbers can be represented by a
graph in the
Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
whose
domain is the entire real line. A more mathematically rigorous definition is given below.
Continuity of real functions is usually defined in terms of
limits. A function with variable is ''continuous at'' the
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 ...
, if the limit of
as tends to , is equal to
There are several different definitions of the (global) continuity of a function, which depend on the nature of its
domain.
A function is continuous on an
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
if the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval
(the whole
real line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
) is often called simply a continuous function; one also says that such a function is ''continuous everywhere''. For example, all
polynomial functions are continuous everywhere.
A function is continuous on a
semi-open or a
closed interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function
is continuous on its whole domain, which is the closed interval
Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some
isolated points. Examples include the reciprocal function
and the tangent function
When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous.
A partial function is ''discontinuous'' at a point if the point belongs to the
topological closure of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions
and
are discontinuous at , and remain discontinuous whichever value is chosen for defining them at . A point where a function is discontinuous is called a ''discontinuity''.
Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above.
Let
be a function whose
domain is contained in
of real numbers.
Some (but not all) possibilities for
are:
*
is the whole
real line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...
; that is,
*
is a
closed interval of the form
where and are real numbers
*
is an
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
of the form
where and are real numbers
In the case of an open interval,
and
do not belong to
, and the values
and
are not defined, and if they are, they do not matter for continuity on
.
Definition in terms of limits of functions
The function is ''continuous at some point'' of its domain if the
limit of
as ''x'' approaches ''c'' through the domain of ''f'', exists and is equal to
In mathematical notation, this is written as
In detail this means three conditions: first, has to be defined at (guaranteed by the requirement that is in the domain of ). Second, the limit of that equation has to exist. Third, the value of this limit must equal
(Here, we have assumed that the domain of ''f'' does not have any
isolated points.)
Definition in terms of neighborhoods
A
neighborhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
of a point ''c'' is a set that contains, at least, all points within some fixed distance of ''c''. Intuitively, a function is continuous at a point ''c'' if the range of ''f'' over the neighborhood of ''c'' shrinks to a single point
as the width of the neighborhood around ''c'' shrinks to zero. More precisely, a function ''f'' is continuous at a point ''c'' of its domain if, for any neighborhood
there is a neighborhood
in its domain such that
whenever
As neighborhoods are defined in any
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 ...
, this definition of a continuous function applies not only for real functions but also when the domain and the
codomain are
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 ...
s and is thus the most general definition. It follows that a function is automatically continuous at every
isolated point of its domain. For example, every real-valued function on the integers is continuous.
Definition in terms of limits of sequences

One can instead require that for any
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 ...
of points in the domain which
converges to ''c'', the corresponding sequence
converges to
In mathematical notation,
Weierstrass and Jordan definitions (epsilon–delta) of continuous functions

Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function
as above and an element
of the domain
,
is said to be continuous at the point
when the following holds: For any positive real number
however small, there exists some positive real number
such that for all
in the domain of
with
the value of
satisfies
Alternatively written, continuity of
at
means that for every
there exists a
such that for all
:
More intuitively, we can say that if we want to get all the
values to stay in some small
neighborhood
A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neigh ...
around
we need to choose a small enough neighborhood for the
values around
If we can do that no matter how small the
neighborhood is, then
is continuous at
In modern terms, this is generalized by the definition of continuity of a function with respect to a
basis for the topology, here the
metric topology.
Weierstrass had required that the interval
be entirely within the domain
, but Jordan removed that restriction.
Definition in terms of control of the remainder
In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity.
A function
Definition using oscillation

Continuity can also be defined in terms of
oscillation: a function ''f'' is continuous at a point
x_0 if and only if its oscillation at that point is zero; in symbols,
\omega_f(x_0) = 0. A benefit of this definition is that it discontinuity: the oscillation gives how the function is discontinuous at a point.
This definition is helpful 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
\varepsilon (hence a
G_ set) – and gives a rapid proof of one direction of the
Lebesgue integrability condition.
The oscillation is equivalent to the
\varepsilon-\delta 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
\varepsilon_0 there is no
\delta that satisfies the
\varepsilon-\delta definition, then the oscillation is at least
\varepsilon_0, and conversely if for every
\varepsilon there is a desired
\delta, the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to 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 ...
.
Definition using the hyperreals
Cauchy defined the continuity of a function in the following intuitive terms: an
infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see ''Cours d'analyse'', page 34).
Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the
hyperreal numbers. In nonstandard analysis, continuity can be defined as follows.
(see
microcontinuity). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to
Augustin-Louis Cauchy's definition of continuity.
Rules for continuity

Proving the continuity of a function by a direct application of the definition is generaly a noneasy task. Fortunately, in practice, most functions are built from simpler functions, and their continuity can be deduced immediately from the way they are defined, by applying the following rules:
* Every
constant function is continuous
* The
identity function is continuous
* ''Addition and multiplication:'' If the functions and are continuous on their respective domains and , then their sum and their product are continuous on the
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
, where and are defined by and .
* ''
Reciprocal:'' If the function is continuous on the domain , then its reciprocal , defined by is continuous on the domain , that is, the domain from which the points such that are removed.
* ''
Function composition:'' If the functions and are continuous on their respective domains and , then the composition defined by is continuous on , that the part of that is mapped by inside .
* The
sine and cosine functions ( and ) are continuous everywhere.
* The
exponential function is continuous everywhere.
* The
natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
is continuous on the domain formed by all positive real numbers .

These rules imply that every
polynomial function is continuous everywhere and that a
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
is continuous everywhere where it is defined, if the numerator and the denominator have no common
zeros. More generally, the quotient of two continuous functions is continuous outside the zeros of the denominator.

An example of a function for which the above rules are not sufficirent is the
sinc function, which is defined by and for . The above rules show immediately that the function is continuous for , but, for proving the continuity at , one has to prove
\lim_ \frac = 1.
As this is true, one gets that the sinc function is continuous function on all real numbers.
Examples of discontinuous functions

An example of a discontinuous function is the
Heaviside step function H, defined by
H(x) = \begin
1 & \text x \ge 0\\
0 & \text x < 0
\end
Pick for instance
\varepsilon = 1/2. Then there is no around
x = 0, i.e. no open interval
(-\delta,\;\delta) with
\delta > 0, that will force all the
H(x) values to be within the of
H(0), i.e. within
(1/2,\;3/2). Intuitively, we can think of this type of discontinuity as a sudden
jump in function values.
Similarly, the
signum or sign function
\sgn(x) = \begin
\;\;\ 1 & \textx > 0\\
\;\;\ 0 & \textx = 0\\
-1 & \textx < 0
\end
is discontinuous at
x = 0 but continuous everywhere else. Yet another example: the function
f(x) = \begin
\sin\left(x^\right)&\textx \neq 0\\
0&\textx = 0
\end
is continuous everywhere apart from
x = 0.
Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined
pathological, for example,
Thomae's function,
f(x)=\begin
1 &\text x=0\\
\frac&\text x = \frac \text\\
0&\textx\text.
\end
is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein,
Dirichlet's function, the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
for the set of rational numbers,
D(x)=\begin
0&\textx\text (\in \R \setminus \Q)\\
1&\textx\text (\in \Q)
\end
is nowhere continuous.
Properties
A useful lemma
Let
f(x) be a function that is continuous at a point
x_0, and
y_0 be a value such
f\left(x_0\right)\neq y_0. Then
f(x)\neq y_0 throughout some neighbourhood of
x_0.
''Proof:'' By the definition of continuity, take
\varepsilon =\frac>0 , then there exists
\delta>0 such that
\left, f(x)-f(x_0)\ < \frac \quad \text \quad , x-x_0, < \delta
Suppose there is a point in the neighbourhood
, x-x_0, <\delta for which
f(x)=y_0; then we have the contradiction
\left, f(x_0)-y_0\ < \frac.
Intermediate value theorem
The
intermediate value theorem is an
existence theorem, based on the real number property of
completeness, and states:
:If the real-valued function ''f'' is continuous on the
closed interval , b and ''k'' is some number between
f(a) and
f(b), then there is some number
c \in , b such that
f(c) = k.
For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.
As a consequence, if ''f'' is continuous on
, b/math> and f(a) and f(b) differ in sign, then, at some point c \in , b f(c) must equal zero.
Extreme value theorem
The
extreme value theorem states that if a function ''f'' is defined on a closed interval
, b/math> (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists c \in , b/math> with f(c) \geq f(x) for all x \in , b The same is true of the minimum of ''f''. These statements are not, in general, true if the function is defined on an open interval (a, b) (or any set that is not both closed and bounded), as, for example, the continuous function f(x) = \frac, defined on the open interval (0,1), does not attain a maximum, being unbounded above.
Relation to differentiability and integrability
Every
differentiable function
f : (a, b) \to \R
is continuous, as can be shown. The
converse does not hold: for example, the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
function
:
f(x)=, x, = \begin
\;\;\ x & \textx \geq 0\\
-x & \textx < 0
\end
is everywhere continuous. However, it is not differentiable at
x = 0 (but is so everywhere else).
Weierstrass's function is also everywhere continuous but nowhere differentiable.
The
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
''f′''(''x'') of a differentiable function ''f''(''x'') need not be continuous. If ''f′''(''x'') is continuous, ''f''(''x'') is said to be ''continuously differentiable''. The set of such functions is denoted
C^1((a, b)). More generally, the set of functions
f : \Omega \to \R
(from an open interval (or
open subset of
\R)
\Omega to the reals) such that ''f'' is
n times differentiable and such that the
n-th derivative of ''f'' is continuous is denoted
C^n(\Omega). See
differentiability class. In the field of computer graphics, properties related (but not identical) to
C^0, C^1, C^2 are sometimes called
G^0 (continuity of position),
G^1 (continuity of tangency), and
G^2 (continuity of curvature); see
Smoothness of curves and surfaces.
Every continuous function
f : , b\to \R
is
integrable (for example in the sense of the
Riemann integral
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 Gö ...
). The converse does not hold, as the (integrable but discontinuous)
sign function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zer ...
shows.
Pointwise and uniform limits

Given 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 ...
f_1, f_2, \dotsc : I \to \R
of functions such that the limit
f(x) := \lim_ f_n(x)
exists for all
x \in D,, the resulting function
f(x) is referred to as the
pointwise limit of the sequence of functions
\left(f_n\right)_. The pointwise limit function need not be continuous, even if all functions
f_n are continuous, as the animation at the right shows. However, ''f'' is continuous if all functions
f_n are continuous and the sequence
converges uniformly, by the
uniform convergence theorem. This theorem can be used to show that the
exponential functions,
logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
s,
square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
function, and
trigonometric functions are continuous.
Directional Continuity
Image:Right-continuous.svg, A right-continuous function
Image:Left-continuous.svg, A left-continuous function
Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and
semi-continuity. Roughly speaking, a function is if no jump occurs when the limit point is approached from the right. Formally, ''f'' is said to be right-continuous at the point ''c'' if the following holds: For any number
\varepsilon > 0 however small, there exists some number
\delta > 0 such that for all ''x'' in the domain with
c < x < c + \delta, the value of
f(x) will satisfy
, f(x) - f(c), < \varepsilon.
This is the same condition as continuous functions, except it is required to hold for ''x'' strictly larger than ''c'' only. Requiring it instead for all ''x'' with
c - \delta < x < c yields the notion of functions. A function is continuous if and only if it is both right-continuous and left-continuous.
Semicontinuity
A function ''f'' is if, roughly, any jumps that might occur only go down, but not up. That is, for any
\varepsilon > 0, there exists some number
\delta > 0 such that for all ''x'' in the domain with
, x - c, < \delta, the value of
f(x) satisfies
f(x) \geq f(c) - \epsilon.
The reverse condition is .
Continuous functions between metric spaces
The concept of continuous real-valued functions can be generalized to functions between
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 ...
s. A metric space is a set
X equipped with a function (called
metric)
d_X, that can be thought of as a measurement of the distance of any two elements in ''X''. Formally, the metric is a function
d_X : X \times X \to \R
that satisfies a number of requirements, notably the
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of Degeneracy (mathematics)#T ...
. Given two metric spaces
\left(X, d_X\right) and
\left(Y, d_Y\right) and a function
f : X \to Y
then
f is continuous at the point
c \in X (with respect to the given metrics) if for any positive real number
\varepsilon > 0, there exists a positive real number
\delta > 0 such that all
x \in X satisfying
d_X(x, c) < \delta will also satisfy
d_Y(f(x), f(c)) < \varepsilon. As in the case of real functions above, this is equivalent to the condition that for every sequence
\left(x_n\right) in
X with limit
\lim x_n = c, we have
\lim f\left(x_n\right) = f(c). The latter condition can be weakened as follows:
f is continuous at the point
c if and only if for every convergent sequence
\left(x_n\right) in
X with limit
c, the sequence
\left(f\left(x_n\right)\right) is a
Cauchy sequence, and
c is in the domain of
f.
The set of points at which a function between metric spaces is continuous is a
G_ set – this follows from the
\varepsilon-\delta definition of continuity.
This notion of continuity is applied, for example, 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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
. A key statement in this area says that a
linear operator
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 pr ...
T : V \to W
between
normed vector spaces
V and
W (which are
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s equipped with a compatible
norm, denoted
\, x\, ) is continuous if and only if it is
bounded, that is, there is a constant
K such that
\, T(x)\, \leq K \, x\,
for all
x \in V.
Uniform, Hölder and Lipschitz continuity

The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way
\delta depends on
\varepsilon and ''c'' in the definition above. Intuitively, a function ''f'' as above is
uniformly continuous if the
\delta does
not depend on the point ''c''. More precisely, it is required that 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
\varepsilon > 0 there exists
\delta > 0 such that for every
c, b \in X with
d_X(b, c) < \delta, we have that
d_Y(f(b), f(c)) < \varepsilon. Thus, any uniformly continuous function is continuous. The converse does not generally hold but holds when the domain space ''X'' is
compact. Uniformly continuous maps can be defined in the more general situation of
uniform space
In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform con ...
s.
A function is
Hölder continuous with exponent α (a real number) if there is a constant ''K'' such that for all
b, c \in X, the inequality
d_Y (f(b), f(c)) \leq K \cdot (d_X (b, c))^\alpha
holds. Any Hölder continuous function is uniformly continuous. The particular case
\alpha = 1 is referred to as
Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant ''K'' such that the inequality
d_Y (f(b), f(c)) \leq K \cdot d_X (b, c)
holds for any
b, c \in X. The Lipschitz condition occurs, for example, in the
Picard–Lindelöf theorem concerning the solutions of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s.
Continuous functions between topological spaces
Another, more abstract, notion of continuity is the continuity of functions between
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 ...
s in which there generally is no formal notion of distance, as there is in the case of
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 ...
s. A topological space is a set ''X'' together with a topology on ''X'', which is a set of
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of ''X'' satisfying a few requirements with respect to their unions and intersections that generalize the properties of the
open balls in metric spaces while still allowing one to talk about the
neighborhoods of a given point. The elements of a topology are called
open subsets of ''X'' (with respect to the topology).
A function
f : X \to Y
between two topological spaces ''X'' and ''Y'' is continuous if for every open set
V \subseteq Y, the
inverse image
f^(V) = \
is an open subset of ''X''. That is, ''f'' is a function between the sets ''X'' and ''Y'' (not on the elements of the topology
T_X), but the continuity of ''f'' depends on the topologies used on ''X'' and ''Y''.
This is equivalent to the condition that the
preimages of the
closed set
In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. In a topological space, a closed set can be defined as a set which contains all its lim ...
s (which are the complements of the open subsets) in ''Y'' are closed in ''X''.
An extreme example: if a set ''X'' is given the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
(in which every subset is open), all functions
f : X \to T
to any topological space ''T'' are continuous. On the other hand, if ''X'' is equipped with the
indiscrete topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
(in which the only open subsets are the empty set and ''X'') and the space ''T'' set is at least
T0, then the only continuous functions are the constant functions. Conversely, any function whose codomain is indiscrete is continuous.
Continuity at a point
The translation in the language of neighborhoods of the
(\varepsilon, \delta)-definition of continuity leads to the following definition of the continuity at a point:
This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using
preimages rather than images.
Also, as every set that contains a neighborhood is also a neighborhood, and
f^(V) is the largest subset of such that
f(U) \subseteq V, this definition may be simplified into:
As an open set is a set that is a neighborhood of all its points, a function
f : X \to Y is continuous at every point of if and only if it is a continuous function.
If ''X'' and ''Y'' are metric spaces, it is equivalent to consider the
neighborhood system of
open balls centered at ''x'' and ''f''(''x'') instead of all neighborhoods. This gives back the above
\varepsilon-\delta definition of continuity in the context of metric spaces. In general topological spaces, there is no notion of nearness or distance. If, however, the target space is a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
, it is still true that ''f'' is continuous at ''a'' if and only if the limit of ''f'' as ''x'' approaches ''a'' is ''f''(''a''). At an isolated point, every function is continuous.
Given
x \in X, a map
f : X \to Y is continuous at
x if and only if whenever
\mathcal is a filter on
X that
converges to
x in
X, which is expressed by writing
\mathcal \to x, then necessarily
f(\mathcal) \to f(x) in
Y.
If
\mathcal(x) denotes the
neighborhood filter at
x then
f : X \to Y is continuous at
x if and only if
f(\mathcal(x)) \to f(x) in
Y. Moreover, this happens if and only if the
prefilter f(\mathcal(x)) is a
filter base for the neighborhood filter of
f(x) in
Y.
Alternative definitions
Several
equivalent definitions for a topological structure exist; thus, several equivalent ways exist to define a continuous function.
Sequences and nets
In several contexts, the topology of a space is conveniently specified in terms of
limit points. This is often accomplished by specifying when a point is the
limit of a sequence
As the positive integer n becomes larger and larger, the value n\times \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n \times \sin\left(\tfrac1\right) equals 1."
In mathematics, the li ...
. Still, for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points
indexed by a
directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a preordered set in which every finite subset has an upper bound. In other words, it is a non-empty preordered set A such that for any a and b in A there exists c in A wit ...
, known as
nets. A function is (Heine-)continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition.
In detail, a function
f : X \to Y is
sequentially continuous if whenever a sequence
\left(x_n\right) in
X converges to a limit
x, the sequence
\left(f\left(x_n\right)\right) converges to
f(x). Thus, sequentially continuous functions "preserve sequential limits." Every continuous function is sequentially continuous. If
X is a
first-countable space and
countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if
X is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called
sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve the limits of nets, and this property characterizes continuous functions.
For instance, consider the case of real-valued functions of one real variable:
''Proof.'' Assume that
f : A \subseteq \R \to \R is continuous at
x_0 (in the sense of
\epsilon-\delta continuity). Let
\left(x_n\right)_ be a sequence converging at
x_0 (such a sequence always exists, for example,
x_n = x, \text n); since
f is continuous at
x_0
\forall \epsilon > 0\, \exists \delta_ > 0 : 0 < , x-x_0, < \delta_ \implies , f(x)-f(x_0), < \epsilon.\quad (*)
For any such
\delta_ we can find a natural number
\nu_ > 0 such that for all
n > \nu_,
, x_n-x_0, < \delta_,
since
\left(x_n\right) converges at
x_0; combining this with
(*) we obtain
\forall \epsilon > 0 \,\exists \nu_ > 0 : \forall n > \nu_ \quad , f(x_n)-f(x_0), < \epsilon.
Assume on the contrary that
f is sequentially continuous and proceed by contradiction: suppose
f is not continuous at
x_0
\exists \epsilon > 0 : \forall \delta_ > 0,\,\exists x_: 0 < , x_-x_0, < \delta_\epsilon \implies , f(x_)-f(x_0), > \epsilon
then we can take
\delta_=1/n,\,\forall n > 0 and call the corresponding point
x_ =: x_n: in this way we have defined a sequence
(x_n)_ such that
\forall n > 0 \quad , x_n-x_0, < \frac,\quad , f(x_n)-f(x_0), > \epsilon
by construction
x_n \to x_0 but
f(x_n) \not\to f(x_0), which contradicts the hypothesis of sequential continuity.
\blacksquare
Closure operator and interior operator definitions
In terms of the
interior and
closure operators, we have the following equivalences,
''Proof.''i ⇒ ii.
Fix a subset
B of
Y. Since
\operatorname_Y B is open.
and
f is continuous,
f^(\operatorname_Y B) is open in
X.
As
\operatorname_Y B \subseteq B, we have
f^(\operatorname_Y B) \subseteq f^(B).
By the definition of the interior,
\operatorname_X\left(f^(B)\right) is the largest open set contained in
f^(B). Hence
f^(\operatorname_Y B) \subseteq \operatorname_X\left(f^(B)\right).
ii ⇒ iii.
Fix
A\subseteq X and let
x\in\operatorname_X A. Suppose to the contrary that
f(x)\notin\operatorname_Y\left(f(A)\right),
then we may find some open neighbourhood
V of
f(x) that is disjoint from
\operatorname_Y\left(f(A)\right). By ii,
f^(V) = f^(\operatorname_Y V) \subseteq \operatorname_X \left(f^(V)\right), hence
f^(V) is open. Then we have found an open neighbourhood of
x that does not intersect
\operatorname_X A, contradicting the fact that
x\in\operatorname_X A.
Hence
f\left(\operatorname_X A\right) \subseteq \operatorname_Y \left(f(A)\right).
iii ⇒ i.
Let
N\subseteq Y be closed. Let
M = f^(N) be the preimage of
N.
By iii, we have
f\left(\operatorname_X M\right) \subseteq \operatorname_Y \left(f(M)\right).
Since
f(M) = f(f^(N)) \subseteq N,
we have further that
f\left(\operatorname_X M\right) \subseteq \operatorname_Y N = N.
Thus
\operatorname_X M \subseteq f^\left(f(\operatorname_X M)\right) \subseteq f^(N) = M.
Hence
M is closed and we are done.
If we declare that a point
x is a subset
A \subseteq X if
x \in \operatorname_X A, then this terminology allows for a
plain English description of continuity:
f is continuous if and only if for every subset
A \subseteq X, f maps points that are close to
A to points that are close to
f(A). Similarly,
f is continuous at a fixed given point
x \in X if and only if whenever
x is close to a subset
A \subseteq X, then
f(x) is close to
f(A).
Instead of specifying topological spaces by their
open subsets, any topology on
X can
alternatively be determined by a
closure operator or by an
interior operator.
Specifically, the map that sends a subset
A of a topological space
X to its
topological closure \operatorname_X A satisfies the
Kuratowski closure axioms. Conversely, for any
closure operator A \mapsto \operatorname A there exists a unique topology
\tau on
X (specifically,
\tau := \) such that for every subset
A \subseteq X, \operatorname A is equal to the topological closure
\operatorname_ A of
A in
(X, \tau). If the sets
X and
Y are each associated with closure operators (both denoted by
\operatorname) then a map
f : X \to Y is continuous if and only if
f(\operatorname A) \subseteq \operatorname (f(A)) for every subset
A \subseteq X.
Similarly, the map that sends a subset
A of
X to its
topological interior \operatorname_X A defines an
interior operator. Conversely, any interior operator
A \mapsto \operatorname A induces a unique topology
\tau on
X (specifically,
\tau := \) such that for every
A \subseteq X, \operatorname A is equal to the topological interior
\operatorname_ A of
A in
(X, \tau). If the sets
X and
Y are each associated with interior operators (both denoted by
\operatorname) then a map
f : X \to Y is continuous if and only if
f^(\operatorname B) \subseteq \operatorname\left(f^(B)\right) for every subset
B \subseteq Y.
Filters and prefilters
Continuity can also be characterized in terms of
filters. A function
f : X \to Y is continuous if and only if whenever a filter
\mathcal on
X converges in
X to a point
x \in X, then the
prefilter f(\mathcal) converges in
Y to
f(x). This characterization remains true if the word "filter" is replaced by "prefilter."
Properties
If
f : X \to Y and
g : Y \to Z are continuous, then so is the composition
g \circ f : X \to Z. If
f : X \to Y is continuous and
* ''X'' is
compact, then ''f''(''X'') is compact.
* ''X'' is
connected, then ''f''(''X'') is connected.
* ''X'' is
path-connected, then ''f''(''X'') is path-connected.
* ''X'' is
Lindelöf, then ''f''(''X'') is Lindelöf.
* ''X'' is
separable, then ''f''(''X'') is separable.
The possible topologies on a fixed set ''X'' are
partially ordered: a topology
\tau_1 is said to be
coarser than another topology
\tau_2 (notation:
\tau_1 \subseteq \tau_2) if every open subset with respect to
\tau_1 is also open with respect to
\tau_2. Then, the
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...
\operatorname_X : \left(X, \tau_2\right) \to \left(X, \tau_1\right)
is continuous if and only if
\tau_1 \subseteq \tau_2 (see also
comparison of topologies
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
Definition
A topology on a set may be defined as t ...
). More generally, a continuous function
\left(X, \tau_X\right) \to \left(Y, \tau_Y\right)
stays continuous if the topology
\tau_Y is replaced by a
coarser topology and/or
\tau_X is replaced by a
finer topology.
Homeomorphisms
Symmetric to the concept of a continuous map is an
open map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...
, for which of open sets are open. If an open map ''f'' has an
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon ...
, that inverse is continuous, and if a continuous map ''g'' has an inverse, that inverse is open. Given a
bijective function ''f'' between two topological spaces, the inverse function
f^ need not be continuous. A bijective continuous function with a continuous inverse function is called a .
If a continuous bijection has as its
domain a
compact 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. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
and its codomain is
Hausdorff, then it is a homeomorphism.
Defining topologies via continuous functions
Given a function
f : X \to S,
where ''X'' is a topological space and ''S'' is a set (without a specified topology), the
final topology on ''S'' is defined by letting the open sets of ''S'' be those subsets ''A'' of ''S'' for which
f^(A) is open in ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is
coarser than the final topology on ''S''. Thus, the final topology is the finest topology on ''S'' that makes ''f'' continuous. If ''f'' is
surjective, this topology is canonically identified with the
quotient topology under the
equivalence relation defined by ''f''.
Dually, for a function ''f'' from a set ''S'' to a topological space ''X'', the
initial topology on ''S'' is defined by designating as an open set every subset ''A'' of ''S'' such that
A = f^(U) for some open subset ''U'' of ''X''. If ''S'' has an existing topology, ''f'' is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on ''S''. Thus, the initial topology is the coarsest topology on ''S'' that makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space (''X'', ''𝜏'') is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''𝜏'' called the subspace topology (or the relative topology ...
of ''S'', viewed as a subset of ''X''.
A topology on a set ''S'' is uniquely determined by the class of all continuous functions
S \to X into all topological spaces ''X''.
Dually, a similar idea can be applied to maps
X \to S.
Related notions
If
f : S \to Y is a continuous function from some subset
S of a topological space
X then a of
f to
X is any continuous function
F : X \to Y such that
F(s) = f(s) for every
s \in S, which is a condition that often written as
f = F\big\vert_S. In words, it is any continuous function
F : X \to Y that
restricts to
f on
S. This notion is used, for example, in the
Tietze extension theorem
In topology, the Tietze extension theorem (also known as the Tietze– Urysohn– Brouwer extension theorem or Urysohn-Brouwer lemma) states that any real-valued, continuous function on a closed subset of a normal topological space
In mathe ...
and the
Hahn–Banach theorem
In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient ...
. If
f : S \to Y is not continuous, then it could not possibly have a continuous extension. If
Y is a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
and
S is a
dense subset of
X then a continuous extension of
f : S \to Y to
X, if one exists, will be unique. The
Blumberg theorem states that if
f : \R \to \R is an arbitrary function then there exists a dense subset
D of
\R such that the restriction
f\big\vert_D : D \to \R is continuous; in other words, every function
\R \to \R can be restricted to some dense subset on which it is continuous.
Various other mathematical domains use the concept of continuity in different but related meanings. For example, in
order theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
, an order-preserving function
f : X \to Y between particular types of
partially ordered set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
s
X and
Y is continuous if for each
directed subset A of
X, we have
\sup f(A) = f(\sup A). Here
\,\sup\, is the
supremum with respect to the orderings in
X and
Y, respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the
Scott topology.
In
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
F : \mathcal C \to \mathcal D
between two
categories is called if it commutes with small
limits. That is to say,
\varprojlim_ F(C_i) \cong F \left(\varprojlim_ C_i \right)
for any small (that is, indexed by a set
I, as opposed to a
class
Class, Classes, or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used d ...
)
diagram
A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age o ...
of
objects in
\mathcal C.
A is a generalization of metric spaces and posets, which uses the concept of
quantales, and that can be used to unify the notions of metric spaces and
domains.
In
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
, a function
f : E \to \mathbb^k defined on a
Lebesgue measurable set E \subseteq \mathbb^n is called
approximately continuous at a point
x_0 \in E if the
approximate limit of
f at
x_0 exists and equals
f(x_0). This generalizes the notion of continuity by replacing the ordinary limit with the
approximate limit. A fundamental result known as the
Stepanov-Denjoy theorem states that a function is
measurable if and only if it is approximately continuous
almost everywhere.
See also
*
Continuity (mathematics)
In mathematics, the terms continuity, continuous, and continuum are used in a variety of related ways.
Continuity of functions and measures
* Continuous function
* Absolutely continuous function
* Absolute continuity of a measure with respe ...
*
Absolute continuity
*
Approximate continuity
*
Dini continuity
*
Equicontinuity
*
Geometric continuity
*
Parametric continuity
*
Classification of discontinuities
*
Coarse function
In mathematics, coarse functions are Function (mathematics), functions that may appear to be continuous at a distance, but in reality are not necessarily continuous.Chul-Woo Lee and Jared Duke (2007)Coarse Function Value Theorems ''Rose-Hulman Unde ...
*
Continuous function (set theory)
*
Continuous stochastic process
*
Normal function
*
Open and closed maps
*
Piecewise
*
Symmetrically continuous function
*
Direction-preserving function - an analog of a continuous function in discrete spaces.
References
Bibliography
*
*
{{Authority control
Calculus
Types of functions