In

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 $\backslash varepsilon\; >\; 0$ however small, there exists some number $\backslash delta\; >\; 0$ such that for all ''x'' in the domain with $c\; <\; x\; <\; c\; +\; \backslash delta,$ the value of $f(x)$ will satisfy
$$,\; f(x)\; -\; f(c),\; <\; \backslash varepsilon.$$
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 $c\; -\; \backslash delta\; <\; x\; <\; c$ 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 $\backslash varepsilon\; >\; 0,$ there exists some number $\backslash delta\; >\; 0$ such that for all ''x'' in the domain with $,\; x\; -\; c,\; <\; \backslash delta,$ the value of $f(x)$ satisfies
$$f(x)\; \backslash geq\; f(c)\; -\; \backslash epsilon.$$
The reverse condition is .

_{0}, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.

domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...

s.

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 induces a continuous variation of the value
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value beyo ...

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 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, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 generalizati ...

and mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...

, 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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

.
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 byBernard Bolzano
Bernard Bolzano (, ; ; ; born Bernardus Placidus Johann Gonzal Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his li ...

in 1817. Augustin-Louis Cauchy defined continuity of $y\; =\; f(x)$ as follows: an infinitely small increment $\backslash alpha$ of the independent variable ''x'' always produces an infinitely small change $f(x+\backslash alpha)-f(x)$ 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 fromreal number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s 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 whose domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...

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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

, if the limit of $f(x),$ as tends to , is equal to $f(c).$
There are several different definitions of (global) continuity of a function, which depend on the nature of its domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...

.
A function is continuous on an open interval 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 $(-\backslash infty,\; +\backslash infty)$ (the whole real line) is often called simply a continuous function; one says also 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
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 $f(x)\; =\; \backslash sqrt$ is continuous on its whole domain, which is the closed interval $;\; href="/html/ALL/l/,+\backslash infty).$
Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some isolated points. Examples are the functions $x\; \backslash mapsto\; \backslash frac$ and $x\backslash mapsto\; \backslash tan\; x.$ 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 $x\backslash mapsto\; \backslash frac$ and $x\backslash mapsto\; \backslash sin(\backslash frac\; )$ 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 $$f\; :\; D\; \backslash to\; \backslash R$$ be a function defined on a subset
In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset of ...

$D$ of the set $\backslash R$ of real numbers.
This subset $D$ is the domain of . Some possible choices include
*$D\; =\; \backslash R$: i.e., $D$ is the whole set of real numbers), or, for and real numbers,
*$D\; =;\; href="/html/ALL/l/,\_b.html"\; ;"title=",\; b">,\; b$: $D$ is a closed interval, or
*$D\; =\; (a,\; b)\; =\; \backslash $: $D$ is an open interval.
In case of the domain $D$ being defined as an open interval, $a$ and $b$ do not belong to $D$, and the values of $f(a)$ and $f(b)$ do not matter for continuity on $D$.
Definition in terms of limits of functions

The function is ''continuous at some point'' of its domain if thelimit
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 $f(x),$ as ''x'' approaches ''c'' through the domain of ''f'', exists and is equal to $f(c).$ In mathematical notation, this is written as
$$\backslash lim\_\; =\; f(c).$$
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 $f(c).$
(Here, we have assumed that the domain of ''f'' does not have any isolated points.)
Definition in terms of neighborhoods

A neighborhood 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 $f(c)$ 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 $N\_1(f(c))$ there is a neighborhood $N\_2(c)$ in its domain such that $f(x)\; \backslash in\; N\_1(f(c))$ whenever $x\backslash in\; N\_2(c).$ This definition only requires that the domain and the codomain aretopological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...

s 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 anysequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

$(x\_n)\_$ of points in the domain which converges to ''c'', the corresponding sequence $\backslash left(f(x\_n)\backslash right)\_$ converges to $f(c).$ In mathematical notation, $$\backslash forall\; (x\_n)\_\; \backslash subset\; D:\backslash lim\_\; x\_n\; =\; c\; \backslash Rightarrow\; \backslash lim\_\; f(x\_n)\; =\; f(c)\backslash ,.$$
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 $f\; :\; D\; \backslash to\; \backslash mathbb$ as above and an element $x\_0$ of the domain $D$, $f$ is said to be continuous at the point $x\_0$ when the following holds: For any positive real number $\backslash varepsilon\; >\; 0,$ however small, there exists some positive real number $\backslash delta\; >\; 0$ such that for all $x$ in the domain of $f$ with $x\_0\; -\; \backslash delta\; <\; x\; <\; x\_0\; +\; \backslash delta,$ the value of $f(x)$ satisfies $$f\backslash left(x\_0\backslash right)\; -\; \backslash varepsilon\; <\; f(x)\; <\; f(x\_0)\; +\; \backslash varepsilon.$$ Alternatively written, continuity of $f\; :\; D\; \backslash to\; \backslash mathbb$ at $x\_0\; \backslash in\; D$ means that for every $\backslash varepsilon\; >\; 0,$ there exists a $\backslash delta\; >\; 0$ such that for all $x\; \backslash in\; D$: $$\backslash left,\; x\; -\; x\_0\backslash \; <\; \backslash delta\; ~~\backslash text~~\; ,\; f(x)\; -\; f(x\_0),\; <\; \backslash varepsilon.$$ More intuitively, we can say that if we want to get all the $f(x)$ values to stay in some small neighborhood around $f\backslash left(x\_0\backslash right),$ we simply need to choose a small enough neighborhood for the $x$ values around $x\_0.$ If we can do that no matter how small the $f(x\_0)$ neighborhood is, then $f$ is continuous at $x\_0.$ 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 $x\_0\; -\; \backslash delta\; <\; x\; <\; x\_0\; +\; \backslash delta$ be entirely within the domain $D$, 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 $C:;\; href="/html/ALL/l/,\backslash infty)\_\backslash to\_[0,\backslash infty.html"\; ;"title=",\backslash infty)\; \backslash to\; [0,\backslash infty">,\backslash infty)\; \backslash to\; [0,\backslash infty$Definition using oscillation

Continuity can also be defined in terms ofoscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendul ...

: a function ''f'' is continuous at a point $x\_0$ if and only if its oscillation at that point is zero; in symbols, $\backslash omega\_f(x\_0)\; =\; 0.$ 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 to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than $\backslash varepsilon$ (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 G ...

.
The oscillation is equivalent to the $\backslash varepsilon-\backslash delta$ 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 (see limit of a function). For ...

, lim inf) to define oscillation: if (at a given point) for a given $\backslash varepsilon\_0$ there is no $\backslash delta$ that satisfies the $\backslash varepsilon-\backslash delta$ definition, then the oscillation is at least $\backslash varepsilon\_0,$ and conversely if for every $\backslash varepsilon$ there is a desired $\backslash delta,$ 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 together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

.
Definition using the hyperreals

Cauchy defined 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 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'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 $$f,\; g\; \backslash colon\; D\; \backslash to\; \backslash R,$$ then the $$s\; =\; f\; +\; g$$ (defined by $s(x)\; =\; f(x)\; +\; g(x)$ for all $x\backslash in\; D$) is continuous in $D.$ The same holds for the , $$p\; =\; f\; \backslash cdot\; g$$ (defined by $p(x)\; =\; f(x)\; \backslash cdot\; g(x)$ for all $x\; \backslash in\; D$) is continuous in $D.$ Combining the above preservations of continuity and the continuity of constant functions and of theidentity function
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, un ...

$I(x)\; =\; x$ one arrives at the continuity of all polynomial functions 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 $x\; \backslash in\; D$ such that $f(x)\; \backslash neq\; 0$)
is continuous in $D\backslash setminus\; \backslash .$
This implies that, excluding the roots of $g,$ the
$$q\; =\; f\; /\; g$$
(defined by $q(x)\; =\; f(x)/g(x)$ for all $x\; \backslash in\; D$, such that $g(x)\; \backslash neq\; 0$)
is also continuous on $D\backslash setminus\; \backslash $.
For example, the function (pictured)
$$y(x)\; =\; \backslash frac$$
is defined for all real numbers $x\; \backslash neq\; -2$ 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 $F\; :\; \backslash R\; \backslash to\; \backslash R$ that agrees with $y(x)$ for all $x\; \backslash neq\; -2.$
Since the function sine is continuous on all reals, the sinc function $G(x)\; =\; \backslash sin(x)/x,$ is defined and continuous for all real $x\; \backslash neq\; 0.$ 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.,
$$G(0)\; =\; \backslash lim\_\; \backslash frac\; =\; 1.$$
Thus, by setting
:$G(x)\; =\; \backslash begin\; \backslash frac\; x\; \&\; \backslash textx\; \backslash ne\; 0\backslash \backslash \; 1\; \&\; \backslash textx\; =\; 0,\; \backslash end$
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. Given two continuous functions
$$g\; :\; D\_g\; \backslash subseteq\; \backslash R\; \backslash to\; R\_g\; \backslash subseteq\; \backslash R\; \backslash quad\; \backslash text\; \backslash quad\; f\; :\; D\_f\; \backslash subseteq\; \backslash R\; \backslash to\; R\_f\; \backslash subseteq\; D\_g,$$
their composition, denoted as
$c\; =\; g\; \backslash circ\; f\; :\; D\_f\; \backslash to\; \backslash R,$ 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 theHeaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...

$H$, defined by
$$H(x)\; =\; \backslash begin\; 1\; \&\; \backslash text\; x\; \backslash ge\; 0\backslash \backslash \; 0\; \&\; \backslash text\; x\; <\; 0\; \backslash end$$
Pick for instance $\backslash varepsilon\; =\; 1/2$. Then there is no around $x\; =\; 0$, i.e. no open interval $(-\backslash delta,\backslash ;\backslash delta)$ with $\backslash delta\; >\; 0,$ that will force all the $H(x)$ values to be within the of $H(0)$, i.e. within $(1/2,\backslash ;3/2)$. Intuitively we can think of this type of discontinuity as a sudden jump
Jumping is a form of locomotion or movement in which an organism or non-living (e.g., robotic) mechanical system propels itself through the air along a ballistic trajectory.
Jump or Jumping also may refer to:
Places
* Jump, Kentucky or Jump S ...

in function values.
Similarly, the signum or sign function
$$\backslash sgn(x)\; =\; \backslash begin\; \backslash ;\backslash ;\backslash \; 1\; \&\; \backslash textx\; >\; 0\backslash \backslash \; \backslash ;\backslash ;\backslash \; 0\; \&\; \backslash textx\; =\; 0\backslash \backslash \; -1\; \&\; \backslash textx\; <\; 0\; \backslash end$$
is discontinuous at $x\; =\; 0$ but continuous everywhere else. Yet another example: the function
$$f(x)\; =\; \backslash begin\; \backslash sin\backslash left(x^\backslash right)\&\backslash textx\; \backslash neq\; 0\backslash \backslash \; 0\&\backslash textx\; =\; 0\; \backslash end$$
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
Thomae's function is a real-valued function of a real variable that can be defined as:
f(x) =
\begin
\frac &\textx = \tfrac\quad (x \text p \in \mathbb Z \text q \in \mathbb N \text\\
0 &\textx \text
\end
It is named after Carl Jo ...

,
$$f(x)=\backslash begin\; 1\; \&\backslash text\; x=0\backslash \backslash \; \backslash frac\&\backslash text\; x\; =\; \backslash frac\; \backslash text\backslash \backslash \; 0\&\backslash textx\backslash text.\; \backslash end$$
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 for the set of rational numbers,
$$D(x)=\backslash begin\; 0\&\backslash textx\backslash text\; (\backslash in\; \backslash R\; \backslash setminus\; \backslash Q)\backslash \backslash \; 1\&\backslash textx\backslash text\; (\backslash in\; \backslash Q)\; \backslash end$$
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 $f\backslash left(x\_0\backslash right)\backslash neq\; y\_0.$ Then $f(x)\backslash neq\; y\_0$ throughout some neighbourhood of $x\_0.$ ''Proof:'' By the definition of continuity, take $\backslash varepsilon\; =\backslash frac>0$ , then there exists $\backslash delta>0$ such that $$\backslash left,\; f(x)-f(x\_0)\backslash \; <\; \backslash frac\; \backslash quad\; \backslash text\; \backslash quad\; ,\; x-x\_0,\; <\; \backslash delta$$ Suppose there is a point in the neighbourhood $,\; x-x\_0,\; <\backslash delta$ for which $f(x)=y\_0;$ then we have the contradiction $$\backslash left,\; f(x\_0)-y\_0\backslash \; <\; \backslash frac.$$Intermediate value theorem

The intermediate value theorem is anexistence 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 $;\; href="/html/ALL/l/,\_b.html"\; ;"title=",\; b">,\; b$ and ''k'' is some number between $f(a)$ and $f(b),$ then there is some number $c\; \backslash in;\; href="/html/ALL/l/,\_b.html"\; ;"title=",\; b">,\; b$ 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 $;\; href="/html/ALL/l/,\_b.html"\; ;"title=",\; b">,\; b$Extreme value theorem

Theextreme value theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> s ...

states that if a function ''f'' is defined on a closed interval $;\; href="/html/ALL/l/,\_b.html"\; ;"title=",\; b">,\; b$Relation to differentiability and integrability

Everydifferentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...

$$f\; :\; (a,\; b)\; \backslash to\; \backslash R$$
is continuous, as can be shown. The converse does not hold: for example, the absolute value function
:$f(x)=,\; x,\; =\; \backslash begin\; \backslash ;\backslash ;\backslash \; x\; \&\; \backslash textx\; \backslash geq\; 0\backslash \backslash \; -x\; \&\; \backslash textx\; <\; 0\; \backslash end$
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 of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

''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
$$f\; :\; \backslash Omega\; \backslash to\; \backslash R$$
(from an open interval (or open subset of $\backslash R$) $\backslash Omega$ to the reals) such that ''f'' is $n$ times differentiable and such that the $n$-th derivative of ''f'' is continuous is denoted $C^n(\backslash Omega).$ See differentiability class
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...

. 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
$$f\; :;\; href="/html/ALL/l/,\_b.html"\; ;"title=",\; b">,\; b$$
is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the (integrable, but discontinuous) sign function shows.
Pointwise and uniform limits

Given asequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

$$f\_1,\; f\_2,\; \backslash dotsc\; :\; I\; \backslash to\; \backslash R$$
of functions such that the limit
$$f(x)\; :=\; \backslash lim\_\; f\_n(x)$$
exists for all $x\; \backslash in\; D,$, the resulting function $f(x)$ is referred to as the pointwise limit of the sequence of functions $\backslash left(f\_n\backslash right)\_.$ 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 arbitrarily s ...

, 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 function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...

s, logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...

s, square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...

function, and trigonometric functions are continuous.
Directional and semi-continuity

Continuous functions between metric spaces

The concept of continuous real-valued functions can be generalized to functions betweenmetric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

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
$$d\_X\; :\; X\; \backslash times\; X\; \backslash to\; \backslash R$$
that satisfies a number of requirements, notably the triangle inequality. Given two metric spaces $\backslash left(X,\; d\_X\backslash right)$ and $\backslash left(Y,\; d\_Y\backslash right)$ and a function
$$f\; :\; X\; \backslash to\; Y$$
then $f$ is continuous at the point $c\; \backslash in\; X$ (with respect to the given metrics) if for any positive real number $\backslash varepsilon\; >\; 0,$ there exists a positive real number $\backslash delta\; >\; 0$ such that all $x\; \backslash in\; X$ satisfying $d\_X(x,\; c)\; <\; \backslash delta$ will also satisfy $d\_Y(f(x),\; f(c))\; <\; \backslash varepsilon.$ As in the case of real functions above, this is equivalent to the condition that for every sequence $\backslash left(x\_n\backslash right)$ in $X$ with limit $\backslash lim\; x\_n\; =\; c,$ we have $\backslash lim\; f\backslash left(x\_n\backslash right)\; =\; f(c).$ The latter condition can be weakened as follows: $f$ is continuous at the point $c$ if and only if for every convergent sequence $\backslash left(x\_n\backslash right)$ in $X$ with limit $c$, the sequence $\backslash left(f\backslash left(x\_n\backslash right)\backslash right)$ 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 $\backslash varepsilon-\backslash delta$ 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 (e.g. inner product, norm, topology, etc.) and the linear functions defi ...

. A key statement in this area says that a linear operator
$$T\; :\; V\; \backslash to\; W$$
between normed vector spaces $V$ and $W$ (which are vector spaces equipped with a compatible norm, denoted $\backslash ,\; x\backslash ,$) is continuous if and only if it is bounded, that is, there is a constant $K$ such that
$$\backslash ,\; T(x)\backslash ,\; \backslash leq\; K\; \backslash ,\; x\backslash ,$$
for all $x\; \backslash in\; V.$
Uniform, Hölder and Lipschitz continuity

The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way $\backslash delta$ depends on $\backslash varepsilon$ and ''c'' in the definition above. Intuitively, a function ''f'' as above is uniformly continuous if the $\backslash delta$ does not depend on the point ''c''. More precisely, it is required that for everyreal number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

$\backslash varepsilon\; >\; 0$ there exists $\backslash delta\; >\; 0$ such that for every $c,\; b\; \backslash in\; X$ with $d\_X(b,\; c)\; <\; \backslash delta,$ we have that $d\_Y(f(b),\; f(c))\; <\; \backslash varepsilon.$ Thus, any uniformly continuous function is continuous. The converse does not hold in general, but holds when the domain space ''X'' is compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...

. Uniformly continuous maps can be defined in the more general situation of uniform spaces.
A function is Hölder continuous Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
* Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modu ...

with exponent α (a real number) if there is a constant ''K'' such that for all $b,\; c\; \backslash in\; X,$ the inequality
$$d\_Y\; (f(b),\; f(c))\; \backslash leq\; K\; \backslash cdot\; (d\_X\; (b,\; c))^\backslash alpha$$
holds. Any Hölder continuous function is uniformly continuous. The particular case $\backslash alpha\; =\; 1$ is referred to as Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant ''K'' such that the inequality
$$d\_Y\; (f(b),\; f(c))\; \backslash leq\; K\; \backslash cdot\; d\_X\; (b,\; c)$$
holds for any $b,\; c\; \backslash in\; X.$ The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem
In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cau ...

concerning the solutions of ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

s.
Continuous functions between topological spaces

Another, more abstract, notion of continuity is continuity of functions betweentopological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...

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 together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

s. A topological space is a set ''X'' together with a topology on ''X'', which is a set of subset
In mathematics, set ''A'' is a subset of a set ''B'' if all 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 unequal, then ''A'' is a proper subset of ...

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 of a given point. The elements of a topology are called open subsets of ''X'' (with respect to the topology).
A function
$$f\; :\; X\; \backslash to\; Y$$
between two topological spaces ''X'' and ''Y'' is continuous if for every open set $V\; \backslash subseteq\; Y,$ the inverse image
$$f^(V)\; =\; \backslash $$
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 sets (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
$$f\; :\; X\; \backslash to\; T$$
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 TContinuity at a point

The translation in the language of neighborhoods of the $(\backslash varepsilon,\; \backslash delta)$-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 $f(U)\; \backslash subseteq\; V,$ this definition may be simplified into: As an open set is a set that is a neighborhood of all its points, a function $f\; :\; X\; \backslash to\; Y$ 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 $\backslash varepsilon-\backslash delta$ 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 aHausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...

, 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 $x\; \backslash in\; X,$ a map $f\; :\; X\; \backslash to\; Y$ is continuous at $x$ if and only if whenever $\backslash mathcal$ is a filter on $X$ that converges to $x$ in $X,$ which is expressed by writing $\backslash mathcal\; \backslash to\; x,$ then necessarily $f(\backslash mathcal)\; \backslash to\; f(x)$ in $Y.$
If $\backslash mathcal(x)$ denotes the neighborhood filter In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbou ...

at $x$ then $f\; :\; X\; \backslash to\; Y$ is continuous at $x$ if and only if $f(\backslash mathcal(x))\; \backslash to\; f(x)$ in $Y.$ Moreover, this happens if and only if the prefilter $f(\backslash mathcal(x))$ 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 thelimit of a sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1."
In mathematics, the limi ...

, 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 A together with a reflexive and transitive binary relation \,\leq\, (that is, a preorder), with the additional property that every pair of elements has ...

, 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 $f\; :\; X\; \backslash to\; Y$ is sequentially continuous if whenever a sequence $\backslash left(x\_n\backslash right)$ in $X$ converges to a limit $x,$ the sequence $\backslash left(f\backslash left(x\_n\backslash right)\backslash right)$ 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 $f\; :\; A\; \backslash subseteq\; \backslash R\; \backslash to\; \backslash R$ is continuous at $x\_0$ (in the sense of $\backslash epsilon-\backslash delta$ continuity). Let $\backslash left(x\_n\backslash right)\_$ be a sequence converging at $x\_0$ (such a sequence always exists, for example, $x\_n\; =\; x,\; \backslash text\; n$); since $f$ is continuous at $x\_0$
$$\backslash forall\; \backslash epsilon\; >\; 0\backslash ,\; \backslash exists\; \backslash delta\_\; >\; 0\; :\; 0\; <\; ,\; x-x\_0,\; <\; \backslash delta\_\; \backslash implies\; ,\; f(x)-f(x\_0),\; <\; \backslash epsilon.\backslash quad\; (*)$$
For any such $\backslash delta\_$ we can find a natural number $\backslash nu\_\; >\; 0$ such that for all $n\; >\; \backslash nu\_,$
$$,\; x\_n-x\_0,\; <\; \backslash delta\_,$$
since $\backslash left(x\_n\backslash right)$ converges at $x\_0$; combining this with $(*)$ we obtain
$$\backslash forall\; \backslash epsilon\; >\; 0\; \backslash ,\backslash exists\; \backslash nu\_\; >\; 0\; :\; \backslash forall\; n\; >\; \backslash nu\_\; \backslash quad\; ,\; f(x\_n)-f(x\_0),\; <\; \backslash epsilon.$$
Assume on the contrary that $f$ is sequentially continuous and proceed by contradiction: suppose $f$ is not continuous at $x\_0$
$$\backslash exists\; \backslash epsilon\; >\; 0\; :\; \backslash forall\; \backslash delta\_\; >\; 0,\backslash ,\backslash exists\; x\_:\; 0\; <\; ,\; x\_-x\_0,\; <\; \backslash delta\_\backslash epsilon\; \backslash implies\; ,\; f(x\_)-f(x\_0),\; >\; \backslash epsilon$$
then we can take $\backslash delta\_=1/n,\backslash ,\backslash forall\; n\; >\; 0$ and call the corresponding point $x\_\; =:\; x\_n$: in this way we have defined a sequence $(x\_n)\_$ such that
$$\backslash forall\; n\; >\; 0\; \backslash quad\; ,\; x\_n-x\_0,\; <\; \backslash frac,\backslash quad\; ,\; f(x\_n)-f(x\_0),\; >\; \backslash epsilon$$
by construction $x\_n\; \backslash to\; x\_0$ but $f(x\_n)\; \backslash not\backslash to\; f(x\_0)$, which contradicts the hypothesis of sequentially continuity. $\backslash blacksquare$
Closure operator and interior operator definitions

In terms of the interior operator, a function $f\; :\; X\; \backslash to\; Y$ between topological spaces is continuous if and only if for every subset $B\; \backslash subseteq\; Y,$ $$f^\backslash left(\backslash operatorname\_Y\; B\backslash right)\; ~\backslash subseteq~\; \backslash operatorname\_X\backslash left(f^(B)\backslash right).$$ In terms of the closure operator, $f\; :\; X\; \backslash to\; Y$ is continuous if and only if for every subset $A\; \backslash subseteq\; X,$ $$f\backslash left(\backslash operatorname\_X\; A\backslash right)\; ~\backslash subseteq~\; \backslash operatorname\_Y\; (f(A)).$$ That is to say, given any element $x\; \backslash in\; X$ that belongs to the closure of a subset $A\; \backslash subseteq\; X,$ $f(x)$ necessarily belongs to the closure of $f(A)$ in $Y.$ If we declare that a point $x$ is a subset $A\; \backslash subseteq\; X$ if $x\; \backslash in\; \backslash operatorname\_X\; A,$ then this terminology allows for a plain English description of continuity: $f$ is continuous if and only if for every subset $A\; \backslash subseteq\; X,$ $f$ maps points that are close to $A$ to points that are close to $f(A).$ Similarly, $f$ is continuous at a fixed given point $x\; \backslash in\; X$ if and only if whenever $x$ is close to a subset $A\; \backslash subseteq\; X,$ 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 aclosure 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 de ...

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 d ...

.
Specifically, the map that sends a subset $A$ of a topological space $X$ to its topological closure $\backslash operatorname\_X\; A$ satisfies the Kuratowski closure axioms. Conversely, for any closure 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 de ...

$A\; \backslash mapsto\; \backslash operatorname\; A$ there exists a unique topology $\backslash tau$ on $X$ (specifically, $\backslash tau\; :=\; \backslash $) such that for every subset $A\; \backslash subseteq\; X,$ $\backslash operatorname\; A$ is equal to the topological closure $\backslash operatorname\_\; A$ of $A$ in $(X,\; \backslash tau).$ If the sets $X$ and $Y$ are each associated with closure operators (both denoted by $\backslash operatorname$) then a map $f\; :\; X\; \backslash to\; Y$ is continuous if and only if $f(\backslash operatorname\; A)\; \backslash subseteq\; \backslash operatorname\; (f(A))$ for every subset $A\; \backslash subseteq\; X.$
Similarly, the map that sends a subset $A$ of $X$ to its topological interior $\backslash operatorname\_X\; A$ 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 d ...

. Conversely, any interior operator $A\; \backslash mapsto\; \backslash operatorname\; A$ induces a unique topology $\backslash tau$ on $X$ (specifically, $\backslash tau\; :=\; \backslash $) such that for every $A\; \backslash subseteq\; X,$ $\backslash operatorname\; A$ is equal to the topological interior $\backslash operatorname\_\; A$ of $A$ in $(X,\; \backslash tau).$ If the sets $X$ and $Y$ are each associated with interior operators (both denoted by $\backslash operatorname$) then a map $f\; :\; X\; \backslash to\; Y$ is continuous if and only if $f^(\backslash operatorname\; B)\; \backslash subseteq\; \backslash operatorname\backslash left(f^(B)\backslash right)$ for every subset $B\; \backslash subseteq\; Y.$
Filters and prefilters

Continuity can also be characterized in terms offilters
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component that ...

. A function $f\; :\; X\; \backslash to\; Y$ is continuous if and only if whenever a filter $\backslash mathcal$ on $X$ converges in $X$ to a point $x\; \backslash in\; X,$ then the prefilter $f(\backslash mathcal)$ converges in $Y$ to $f(x).$ This characterization remains true if the word "filter" is replaced by "prefilter."
Properties

If $f\; :\; X\; \backslash to\; Y$ and $g\; :\; Y\; \backslash to\; Z$ are continuous, then so is the composition $g\; \backslash circ\; f\; :\; X\; \backslash to\; Z.$ If $f\; :\; X\; \backslash to\; Y$ is continuous and * ''X'' iscompact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...

, 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 $\backslash tau\_1$ is said to be coarser than another topology $\backslash tau\_2$ (notation: $\backslash tau\_1\; \backslash subseteq\; \backslash tau\_2$) if every open subset with respect to $\backslash tau\_1$ is also open with respect to $\backslash tau\_2.$ 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 ...

$$\backslash operatorname\_X\; :\; \backslash left(X,\; \backslash tau\_2\backslash right)\; \backslash to\; \backslash left(X,\; \backslash tau\_1\backslash right)$$
is continuous if and only if $\backslash tau\_1\; \backslash subseteq\; \backslash tau\_2$ (see also comparison of topologies). More generally, a continuous function
$$\backslash left(X,\; \backslash tau\_X\backslash right)\; \backslash to\; \backslash left(Y,\; \backslash tau\_Y\backslash right)$$
stays continuous if the topology $\backslash tau\_Y$ is replaced by a coarser topology and/or $\backslash tau\_X$ is replaced by a finer topology
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 th ...

.
Homeomorphisms

Symmetric to the concept of a continuous map is an open map, 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 itsdomain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...

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 by making precise the idea of a space having no "punctures" or "missing endpoints", i ...

and its codomain is Hausdorff, then it is a homeomorphism.
Defining topologies via continuous functions

Given a function $$f\; :\; X\; \backslash to\; S,$$ 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, 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 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 $S\; \backslash to\; X$ into all topological spaces ''X''. Dually, a similar idea can be applied to maps $X\; \backslash to\; S.$Related notions

If $f\; :\; S\; \backslash to\; Y$ is a continuous function from some subset $S$ of a topological space $X$ then a of $f$ to $X$ is any continuous function $F\; :\; X\; \backslash to\; Y$ such that $F(s)\; =\; f(s)$ for every $s\; \backslash in\; S,$ which is a condition that often written as $f\; =\; F\backslash big\backslash vert\_S.$ In words, it is any continuous function $F\; :\; X\; \backslash to\; Y$ that restricts to $f$ on $S.$ This notion is used, for example, in the Tietze extension theorem and the Hahn–Banach theorem. Were $f\; :\; S\; \backslash to\; Y$ not continuous then it could not possibly have a continuous extension. If $Y$ is aHausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the ma ...

and $S$ is a dense subset of $X$ then a continuous extension of $f\; :\; S\; \backslash to\; Y$ to $X,$ if one exists, will be unique. The Blumberg theorem
In mathematics, the Blumberg theorem states that for any real function f : \R \to \R there is a dense subset D of \mathbb such that the restriction of f to D is continuous.
For instance, the restriction of the Dirichlet function (the indica ...

states that if $f\; :\; \backslash R\; \backslash to\; \backslash R$ is an arbitrary function then there exists a dense subset $D$ of $\backslash R$ such that the restriction $f\backslash big\backslash vert\_D\; :\; D\; \backslash to\; \backslash R$ is continuous; in other words, every function $\backslash R\; \backslash to\; \backslash R$ 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 $f\; :\; X\; \backslash to\; Y$ between particular types of partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...

s $X$ and $Y$ is continuous if for each directed subset $A$ of $X,$ we have $\backslash sup\; f(A)\; =\; f(\backslash sup\; A).$ Here $\backslash ,\backslash sup\backslash ,$ 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 ...

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, a functor
$$F\; :\; \backslash mathcal\; C\; \backslash to\; \backslash mathcal\; D$$
between two categories is called if it commutes with small limits. That is to say,
$$\backslash varprojlim\_\; F(C\_i)\; \backslash cong\; F\; \backslash left(\backslash varprojlim\_\; C\_i\; \backslash right)$$
for any small (that is, indexed by a set $I,$ as opposed to a class
Class 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 differently ...

) diagram of objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...

in $\backslash mathcal\; C$.
A is a generalization of metric spaces and posets, which uses the concept of quantale In mathematics, quantales are certain partially ordered algebraic structures that generalize locales ( point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann ...

s, and that can be used to unify the notions of metric spaces and 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 respec ...

* 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
* Parametric continuity
* Classification of discontinuities
* 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 c ...

* Open and closed maps
* Piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...

* Symmetrically continuous function
* Direction-preserving function - an analogue of a continuous function in discrete spaces.
References

Bibliography

* * {{Authority control Calculus Types of functions