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 continuous function is a
function such that a continuous variation (that is a change without jump) of the
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
induces a continuous variation of the
value of the function. This means that 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 . Up until the 19th century, mathematicians largely relied on
intuitive
Intuition is the ability to acquire knowledge without recourse to conscious reasoning. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledge; unconscious cognitio ...
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 and
mathematical analysis, 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.
A stronger form of continuity is
uniform continuity. In
order theory, 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account at time would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.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
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
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 numbers 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 (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
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 , if the limit of
as tends to , is equal to
There are several different definitions of (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 a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
if the interval is contained in the domain of the function, and the function is continuous at every point of the interval. A function that is continuous on the interval
(the whole
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 ...
) is often called simply a continuous function; one says also that such a function is ''continuous everywhere''. For example, all
polynomial function
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s are continuous everywhere.
A function is continuous on a
semi-open or a
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
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 are the functions
and
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 with their behavior near the exceptional points, one says that 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, there are several ways to define continuous functions in each of the three senses mentioned above.
Let
be a function defined on a
subset
In mathematics, 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 are ...
of the set
of real numbers.
This subset
is the domain of . Some possible choices include
*
: i.e.,
is the whole set of real numbers), or, for and real numbers,
*
, b
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
= \ :
is a
closed interval, or
*
:
is an
open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
.
In case of the domain
being defined as an open interval,
and
do not belong to
, and the values of
and
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
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 ...
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 (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 area, ...
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
This definition only requires that the domain and the
codomain are
topological spaces and is thus the most general definition. It follows from this definition that a function ''f'' is automatically continuous at every
isolated point of its domain. As a specific example, every real valued function on the set of integers is continuous.
Definition in terms of limits of sequences

One can instead require that for any
sequence 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 (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 area, ...
around
we simply 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 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
\varepsilon (hence a
G_ 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 ...
.
The oscillation is equivalent to the
\varepsilon-\delta 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 (mathematics), function (see limi ...
,
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.
Definition using the hyperreals
Cauchy defined continuity of a function in the following intuitive terms: an
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
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 the addition of 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 to an infinitesimal change of the dependent variable, giving a modern expression to
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
's definition of continuity.
Construction of continuous functions

Checking the continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function. It is straightforward to show that the sum of two functions, continuous on some domain, is also continuous on this domain. Given
f, g \colon D \to \R,
then the
s = f + g
(defined by
s(x) = f(x) + g(x) for all
x\in D) is continuous in
D.
The same holds for the ,
p = f \cdot g
(defined by
p(x) = f(x) \cdot g(x) for all
x \in D)
is continuous in
D.
Combining the above preservations of continuity and the continuity of
constant function
In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image).
Basic properties ...
s and of the
identity function I(x) = x one arrives at the continuity of all
polynomial function
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s such as
f(x) = x^3 + x^2 - 5 x + 3
(pictured on the right).

In the same way it can be shown that the
r = 1/f
(defined by
r(x) = 1/f(x) for all
x \in D such that
f(x) \neq 0)
is continuous in
D\setminus \.
This implies that, excluding the roots of
g, the
q = f / g
(defined by
q(x) = f(x)/g(x) for all
x \in D, such that
g(x) \neq 0)
is also continuous on
D\setminus \.
For example, the function (pictured)
y(x) = \frac
is defined for all real numbers
x \neq -2 and is continuous at every such point. Thus it is a continuous function. The question of continuity at
x = -2 does not arise, since
x = -2 is not in the domain of
y. There is no continuous function
F : \R \to \R that agrees with
y(x) for all
x \neq -2.

Since the function
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
is continuous on all reals, the
sinc function
In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized..
In mathematics, the historical unnormalized sinc function is defined for by
\operatornamex = \frac.
Alternatively, the u ...
G(x) = \sin(x)/x, is defined and continuous for all real
x \neq 0. However, unlike the previous example, ''G'' be extended to a continuous function on real numbers, by the value
G(0) to be 1, which is the limit of
G(x), when ''x'' approaches 0, i.e.,
G(0) = \lim_ \frac = 1.
Thus, by setting
:
G(x) =
\begin
\frac x & \textx \ne 0\\
1 & \textx = 0,
\end
the sinc-function becomes a continuous function on all real numbers. The term is used in such cases, when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points.
A more involved construction of continuous functions is the
function composition
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
. Given two continuous functions
g : D_g \subseteq \R \to R_g \subseteq \R \quad \text \quad f : D_f \subseteq \R \to R_f \subseteq D_g,
their composition, denoted as
c = g \circ f : D_f \to \R, and defined by
c(x) = g(f(x)), is continuous.
This construction allows stating, for example, that
e^
is continuous for all
x > 0.
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
In mathematics, the Dirichlet function is the indicator function 1Q or \mathbf_\Q of the set of rational numbers Q, i.e. if ''x'' is a rational number and if ''x'' is not a rational number (i.e. an irrational number).
\mathbf 1_\Q(x) = \begin
1 & ...
, 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 , one has \mathbf_(x)=1 if x\i ...
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
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval.
This has two import ...
is an
existence theorem
In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential ...
, based on the real number property of
completeness, and states:
:If the real-valued function ''f'' is continuous on the
closed interval , b
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
and ''k'' is some number between
f(a) and
f(b), then there is some number
c \in , b
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
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
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> and
f(a) and
f(b) differ in
sign
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
, then, at some point
c \in , b
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
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
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists
c \in , b
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> with
f(c) \geq f(x) for all
x \in , b
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
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 is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
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 ''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
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 suff ...
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
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\to \R
is
integrable (for example in the sense of the
Riemann integral). 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 an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avoi ...
shows.
Pointwise and uniform limits

Given a
sequence
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
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitra ...
, by the
uniform convergence theorem
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
. This theorem can be used to show that the
exponential functions,
logarithms,
square root function, and
trigonometric functions are continuous.
Directional and semi-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 for continuous functions, except that 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.
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 spaces. 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. 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. 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 pre ...
T : V \to W
between
normed vector spaces
V and
W (which are
vector spaces equipped with a compatible
norm, denoted
\, x\, ) is continuous if and only if it is
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
, 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 \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 hold in general, but holds when the domain space ''X'' is
compact. Uniformly continuous maps can be defined in the more general situation of
uniform spaces.
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 equations.
Continuous functions between topological spaces
Another, more abstract, notion of continuity is continuity of functions between
topological spaces in which there generally is no formal notion of distance, as there is in the case of
metric spaces. A topological space is a set ''X'' together with a topology on ''X'', which is a set of
subset
In mathematics, 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 are ...
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 to talk about the
neighbourhoods
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 ar ...
of a given point. The elements of a topology are called
open subset
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 suff ...
s 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 whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
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 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 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 range 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
preimage
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
s 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, 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 and thus there are several equivalent ways to define a continuous function.
Sequences and nets
In several contexts, the topology of a space is conveniently specified in terms of
limit points. In many instances, this is accomplished by specifying when a point is the
limit of a sequence, but 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 nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive binary relation \,\leq\, (that is, a preorder), with ...
, 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
The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function. That is, given a function ''A'' with domain N (where ...
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 limits of nets, and in fact 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 sequentially continuity.
\blacksquare
Closure operator and interior operator definitions
In terms of the
interior
Interior may refer to:
Arts and media
* ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* ''The Interior'' (novel), by Lisa See
* Interior de ...
operator, a function
f : X \to Y between topological spaces is continuous if and only if for every subset
B \subseteq Y,
f^\left(\operatorname_Y B\right) ~\subseteq~ \operatorname_X\left(f^(B)\right).
In terms of the
closure operator,
f : X \to Y is continuous if and only if for every subset
A \subseteq X,
f\left(\operatorname_X A\right) ~\subseteq~ \operatorname_Y (f(A)).
That is to say, given any element
x \in X that belongs to the closure of a subset
A \subseteq X, f(x) necessarily belongs to the closure of
f(A) in
Y. 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 In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S
:
Closure operators are det ...
.
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
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of the ...
\operatorname_X A defines an
interior operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S
:
Closure operators are det ...
. 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
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
, 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
\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). 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, a ...
, for which of open sets are open. In fact, if an open map ''f'' has an
inverse function, 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 continuous inverse function is called a .
If a continuous bijection has as its
domain a
compact space 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 can be characterized as the finest topology on ''S'' that makes ''f'' continuous. If ''f'' is
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
, this topology is canonically identified with the
quotient topology under the
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each 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 can be characterized as the coarsest topology on ''S'' that makes ''f'' continuous. If ''f'' is injective, this topology is canonically identified with the
subspace 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 and the
Hahn–Banach theorem. Were
f : S \to Y not continuous then it could not possibly have a continuous extension. If
Y is a
Hausdorff space 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, an order-preserving function
f : X \to Y between particular types of
partially ordered sets
X and
Y is continuous if for each
directed subset
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has an ...
A of
X, we have
\sup f(A) = f(\sup A). Here
\,\sup\, is 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 l ...
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 that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a
functor
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)
diagram
A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three- ...
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.
See also
*
Continuity (mathematics)
*
Absolute continuity
*
Dini continuity
*
Equicontinuity
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
*
Geometric continuity
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
*
Parametric continuity
*
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 ...
*
Coarse function
*
Continuous function (set theory) In set theory, a continuous function is a sequence of ordinals such that the values assumed at limit stages are the limits ( limit suprema and limit infima) of all values at previous stages. More formally, let γ be an ordinal, and s := \langle s_, ...
*
Continuous stochastic process
*
Normal function
In axiomatic set theory, a function ''f'' : Ord → Ord is called normal (or a normal function) if and only if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two ...
*
Open and closed maps
*
Piecewise
*
Symmetrically continuous function 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 mode ...
*
Direction-preserving function In discrete mathematics, a direction-preserving function (or mapping) is a function on a discrete space, such as the integer grid, that (informally) does not change too drastically between two adjacent points. It can be considered a discrete analog ...
- an analogue of a continuous function in discrete spaces.
References
Bibliography
*
*
{{Authority control
Calculus
Types of functions