TheInfoList

OR:

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 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 cognition ...
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 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 i ...
, where arguments and values of functions are real and
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
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 intro ...
, especially in
domain theory Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer ...
, a related concept of continuity is
Scott continuity In mathematics, given two partially ordered sets ''P'' and ''Q'', a function ''f'': ''P'' → ''Q'' between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subse ...
. 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 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 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 $y = f\left(x\right)$ as follows: an infinitely small increment $\alpha$ of the independent variable ''x'' always produces an infinitely small change $f\left(x+\alpha\right)-f\left(x\right)$ 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 In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or ''S''-continuity) of an internal function ''f'' at a point ''a'' is defined as follows: :for all ''x'' infinitely close to ''a'', the value ''f''(''x'') is i ...
). 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 Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematic ...
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 Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at ...
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 Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
in 1854.

# Real functions

## Definition A
real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an interv ...
, that is a function from real numbers to real numbers, can be represented by a
graph Graph may refer to: Mathematics * Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of disc ...
in the
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
; such a function is continuous if, roughly speaking, the graph is a single unbroken curve 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 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 ...
. A function with variable is ''continuous at'' the real number , if the limit of $f\left(x\right),$ as tends to , is equal to $f\left(c\right).$ 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 $\left(-\infty, +\infty\right)$ (the whole
real line In elementary mathematics, a number line is a picture of a graduated straight 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 number to a po ...
) 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 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 $f\left(x\right) = \sqrt$ 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 point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
s. Examples are the functions $x \mapsto \frac$ and $x\mapsto \tan x.$ 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 In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection ...
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 $x\mapsto \frac$ and $x\mapsto \sin(\frac )$ 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 $f : D \to \R$ be a function defined on a subset $D$ of the set $\R$ of real numbers. This subset $D$ is the domain of . Some possible choices include *$D = \R$: i.e., $D$ is the whole set of real numbers), or, for and real numbers, *$D =$
, 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 ...
= \ : $D$ is a
closed 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 ...
, or *$D = \left(a, b\right) = \$: $D$ 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 $D$ being defined as an open interval, $a$ and $b$ do not belong to $D$, and the values of $f\left(a\right)$ and $f\left(b\right)$ 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 the limit of $f\left(x\right),$ as ''x'' approaches ''c'' through the domain of ''f'', exists and is equal to $f\left(c\right).$ In mathematical notation, this is written as $\lim_ = f(c).$ 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 $f\left(c\right).$ (Here, we have assumed that the domain of ''f'' does not have any
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
s.)

### Definition in terms of neighborhoods

A neighborhood (mathematics), 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 $f\left(c\right)$ 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 $N_1\left(f\left(c\right)\right)$ there is a neighborhood $N_2\left(c\right)$ in its domain such that $f\left(x\right) \in N_1\left(f\left(c\right)\right)$ whenever $x\in N_2\left(c\right).$ This definition only requires that the domain and the
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
are
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s and is thus the most general definition. It follows from this definition that a function ''f'' is automatically continuous at every
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
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 (mathematics), sequence $\left(x_n\right)_$ of points in the domain which converges to ''c'', the corresponding sequence $\left\left(f\left(x_n\right)\right\right)_$ converges to $f\left(c\right).$ In mathematical notation, $\forall (x_n)_ \subset D:\lim_ x_n = c \Rightarrow \lim_ f(x_n) = f(c)\,.$

### 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 $f : D \to \mathbb$ 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 $\varepsilon > 0,$ however small, there exists some positive real number $\delta > 0$ such that for all $x$ in the domain of $f$ with $x_0 - \delta < x < x_0 + \delta,$ the value of $f\left(x\right)$ satisfies $f\left(x_0\right) - \varepsilon < f(x) < f(x_0) + \varepsilon.$ Alternatively written, continuity of $f : D \to \mathbb$ at $x_0 \in D$ means that for every $\varepsilon > 0,$ there exists a $\delta > 0$ such that for all $x \in D$: $\left, x - x_0\ < \delta ~~\text~~ , f(x) - f(x_0), < \varepsilon.$ More intuitively, we can say that if we want to get all the $f\left(x\right)$ 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 $f\left\left(x_0\right\right),$ 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 $f\left(x_0\right)$ 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 $x_0 - \delta < x < x_0 + \delta$ 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

### Definition using oscillation Continuity can also be defined in terms of
oscillation 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 pendulum ...
: a function ''f'' is continuous at a point $x_0$ if and only if its oscillation at that point is zero; in symbols, $\omega_f\left(x_0\right) = 0.$ 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. The oscillation is equivalent to the $\varepsilon-\delta$ definition by a simple re-arrangement, and by using a limit ( lim sup, lim inf) to define oscillation: if (at a given point) for a given $\varepsilon_0$ there is no $\delta$ that satisfies the $\varepsilon-\delta$ definition, then the oscillation is at least $\varepsilon_0,$ and conversely if for every $\varepsilon$ there is a desired $\delta,$ the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a
metric space In mathematics, a metric space is a set 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 sett ...
.

### Definition using the hyperreals

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 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 refer ...
change in the independent variable corresponds to an infinitesimal change of the dependent variable (see ''Cours d'analyse'', page 34).
Non-standard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
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 mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains number ...
. In nonstandard analysis, continuity can be defined as follows. (see
microcontinuity In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or ''S''-continuity) of an internal function ''f'' at a point ''a'' is defined as follows: :for all ''x'' infinitely close to ''a'', the value ''f''(''x'') is i ...
). 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 $s\left(x\right) = f\left(x\right) + g\left(x\right)$ for all $x\in D$) is continuous in $D.$ The same holds for the , $p = f \cdot g$ (defined by $p\left(x\right) = f\left(x\right) \cdot g\left(x\right)$ 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 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, unch ...
$I\left(x\right) = x$ 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 $r\left(x\right) = 1/f\left(x\right)$ for all $x \in D$ such that $f\left(x\right) \neq 0$) is continuous in $D\setminus \.$ This implies that, excluding the roots of $g,$ the $q = f / g$ (defined by $q\left(x\right) = f\left(x\right)/g\left(x\right)$ for all $x \in D$, such that $g\left(x\right) \neq 0$) 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 $y\left(x\right)$ 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 ...
$G\left(x\right) = \sin\left(x\right)/x,$ 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 $G\left(0\right)$ to be 1, which is the limit of $G\left(x\right),$ when ''x'' approaches 0, i.e., $G(0) = \lim_ \frac = 1.$ Thus, by setting :$G\left(x\right) = \begin \frac x & \textx \ne 0\\ 1 & \textx = 0, \end$ 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. 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 $c\left(x\right) = g\left(f\left(x\right)\right),$ 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 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 argume ...
$H$, 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 $\left(-\delta,\;\delta\right)$ with $\delta > 0,$ that will force all the $H\left(x\right)$ values to be within the of $H\left(0\right)$, i.e. within $\left(1/2,\;3/2\right)$. 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 Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in t ...
, for example, Thomae's function, $f(x)=\begin 1 &\text x=0\\ \frac&\text x = \frac \text\\ 0&\textx\text. \end$ is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, Dirichlet's function, the
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , 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 $f\left(x\right)$ be a function that is continuous at a point $x_0,$ and $y_0$ be a value such $f\left\left(x_0\right\right)\neq y_0.$ Then $f\left(x\right)\neq y_0$ 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 $f\left(x\right)=y_0;$ 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 impo ...
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 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 ...

, 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 $f\left(a\right)$ and $f\left(b\right),$ 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 $f\left(c\right) = k.$ 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 $f\left(a\right)$ and $f\left(b\right)$ 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 ...
, 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 ...
$f\left(c\right)$ must equal
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usua ...
.

### Extreme value theorem

The
extreme 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> su ...
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 $f\left(c\right) \geq f\left(x\right)$ 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 $\left(a, b\right)$ (or any set that is not both closed and bounded), as, for example, the continuous function $f\left(x\right) = \frac,$ defined on the open interval (0,1), does not attain a maximum, being unbounded above.

### Relation to differentiability and integrability

Every
differentiable 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 i ...
$f : (a, b) \to \R$ is continuous, as can be shown. The converse does not hold: for example, the absolute value function :$f\left(x\right)=, x, = \begin \;\;\ x & \textx \geq 0\\ -x & \textx < 0 \end$ 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 $C^1\left(\left(a, b\right)\right).$ 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 ...
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 $C^n\left(\Omega\right).$ 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 In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gö ...
). The converse does not hold, as the (integrable, but discontinuous)
sign function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avo ...
shows.

### Pointwise and uniform limits Given a sequence (mathematics), 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 $f\left(x\right)$ is referred to as the pointwise limit of the sequence of functions $\left\left(f_n\right\right)_.$ The pointwise limit function need not be continuous, even if all functions $f_n$ are continuous, as the animation at the right shows. However, ''f'' is continuous if all functions $f_n$ are continuous and the sequence converges uniformly, by the uniform convergence theorem. This theorem can be used to show that the
exponential 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, al ...
s, logarithms, square root function, and
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s 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 In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f is upper (respectively, lower) semicontinuous at a point x_0 if, ro ...
. 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 $f\left(x\right)$ 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 $f\left(x\right)$ satisfies $f(x) \geq f(c) - \epsilon.$ The reverse condition is .

# Continuous functions between metric spaces

The concept of continuous real-valued functions can be generalized to functions between
metric space In mathematics, a metric space is a set 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 sett ...
s. A metric space is a set $X$ equipped with a function (called metric) $d_X,$ that can be thought of as a measurement of the distance of any two elements in ''X''. Formally, the metric is a function $d_X : X \times X \to \R$ that satisfies a number of requirements, notably the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
. Given two metric spaces $\left\left(X, d_X\right\right)$ and $\left\left(Y, d_Y\right\right)$ 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 $d_X\left(x, c\right) < \delta$ will also satisfy $d_Y\left(f\left(x\right), f\left(c\right)\right) < \varepsilon.$ As in the case of real functions above, this is equivalent to the condition that for every sequence $\left\left(x_n\right\right)$ in $X$ with limit $\lim x_n = c,$ we have $\lim f\left\left(x_n\right\right) = f\left(c\right).$ The latter condition can be weakened as follows: $f$ is continuous at the point $c$ if and only if for every convergent sequence $\left\left(x_n\right\right)$ in $X$ with limit $c$, the sequence $\left\left(f\left\left(x_n\right\right)\right\right)$ is a
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
, 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 space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
s $V$ and $W$ (which are
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s equipped with a compatible norm, denoted $\, x\,$) is continuous if and only if it is bounded, that is, there is a constant $K$ such that $\, T(x)\, \leq K \, x\,$ for all $x \in V.$

## Uniform, Hölder and Lipschitz continuity The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way $\delta$ depends on $\varepsilon$ and ''c'' in the definition above. Intuitively, a function ''f'' as above is uniformly continuous if the $\delta$ does not depend on the point ''c''. More precisely, it is required that for every real number $\varepsilon > 0$ there exists $\delta > 0$ such that for every $c, b \in X$ with $d_X\left(b, c\right) < \delta,$ we have that $d_Y\left(f\left(b\right), f\left(c\right)\right) < \varepsilon.$ 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 British ...
. Uniformly continuous maps can be defined in the more general situation of
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uni ...
s. A function is
Hölder continuous Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition In mathematics, a real or complex-valued function ''f'' on ''d''-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real c ...
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 In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
. 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 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 between
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
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 sett ...
s. A topological space is a set ''X'' together with a topology on ''X'', which is a set of subsets 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 area, ...
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 ...
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 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 $f : X \to T$ 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 $\left(\varepsilon, \delta\right)$-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 $f^\left(V\right)$ is the largest subset of such that $f\left(U\right) \subseteq V,$ 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 $f\left(\mathcal\right) \to f\left(x\right)$ in $Y.$ If $\mathcal\left(x\right)$ denotes the neighborhood filter at $x$ then $f : X \to Y$ is continuous at $x$ if and only if $f\left(\mathcal\left(x\right)\right) \to f\left(x\right)$ in $Y.$ Moreover, this happens if and only if the prefilter $f\left(\mathcal\left(x\right)\right)$ is a filter base for the neighborhood filter of $f\left(x\right)$ 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 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 a ...
, 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 In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of co ...
if whenever a sequence $\left\left(x_n\right\right)$ in $X$ converges to a limit $x,$ the sequence $\left\left(f\left\left(x_n\right\right)\right\right)$ converges to $f\left(x\right).$ Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If $X$ is a
first-countable space In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if $X$ is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called
sequential space In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
s.) 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 $\left\left(x_n\right\right)_$ 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 $\left\left(x_n\right\right)$ converges at $x_0$; combining this with $\left(*\right)$ 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 $\left(x_n\right)_$ 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 $f\left(x_n\right) \not\to f\left(x_0\right)$, which contradicts the hypothesis of sequentially continuity. $\blacksquare$

### Closure operator and interior operator definitions

In terms of the interior 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 $A \subseteq X,$ $f\left(x\right)$ necessarily belongs to the closure of $f\left(A\right)$ 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 $f\left(A\right).$ 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 $f\left(x\right)$ is close to $f\left(A\right).$ Instead of specifying topological spaces by their open subsets, any topology on $X$ can alternatively be determined by a closure operator or by an interior operator. Specifically, the map that sends a subset $A$ of a topological space $X$ to its
topological closure In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection ...
$\operatorname_X A$ 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 $\left(X, \tau\right).$ 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 $f\left(\operatorname A\right) \subseteq \operatorname \left(f\left(A\right)\right)$ for every subset $A \subseteq X.$ Similarly, the map that sends a subset $A$ of $X$ to its topological interior $\operatorname_X A$ defines an interior operator. Conversely, any interior operator $A \mapsto \operatorname A$ induces a unique topology $\tau$ on $X$ (specifically, $\tau := \$) such that for every $A \subseteq X,$ $\operatorname A$ is equal to the topological interior $\operatorname_ A$ of $A$ in $\left(X, \tau\right).$ 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 $f^\left(\operatorname B\right) \subseteq \operatorname\left\left(f^\left(B\right)\right\right)$ 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 $f\left(\mathcal\right)$ converges in $Y$ to $f\left(x\right).$ 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 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 British ...
, then ''f''(''X'') is compact. * ''X'' is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
, 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 t ...
, 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 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 ...
: 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 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 the ...
.

## Homeomorphisms

Symmetric to the concept of a continuous map is an
open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
, for which of open sets are open. In fact, if an open map ''f'' has an
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\t ...
, 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 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", ...
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 $f^\left(A\right)$ 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 In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
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 relatio ...
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 $A = f^\left(U\right)$ 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 In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
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 $F\left(s\right) = f\left(s\right)$ 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 The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear ...
. 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 In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of $X$ then a continuous extension of $f : S \to Y$ to $X,$ if one exists, will be unique. The Blumberg theorem states that if $f : \R \to \R$ is an arbitrary function then there exists a dense subset $D$ of $\R$ such that the restriction $f\big\vert_D : D \to \R$ is continuous; in other words, every function $\R \to \R$ can be restricted to some dense subset on which it is continuous. Various other mathematical domains use the concept of continuity in different, but related meanings. For example, in
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intro ...
, an order-preserving function $f : X \to Y$ 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 $\sup f\left(A\right) = f\left(\sup A\right).$ 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 low ...
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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
$F : \mathcal C \to \mathcal D$ between two
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) *Categories (Peirce) ...
is called if it commutes with small
limits 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 ...
. 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
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 a ...
s, and that can be used to unify the notions of metric spaces and domains.

* Continuity (mathematics) *
Absolute continuity In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central op ...
* Dini continuity * Equicontinuity * Geometric continuity *
Parametric 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 ...
*
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 o ...
* Coarse function * Continuous function (set theory) * Continuous stochastic process * Normal function * 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. Pi ...
* Symmetrically continuous function * Direction-preserving function - an analogue of a continuous function in discrete spaces.

# Bibliography

* * {{Authority control Calculus Types of functions