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mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to sufficiently small changes in its input. If not continuous, a function is said to be ''discontinuous''. Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, during which attempts such as the epsilon–delta definition were made to formalize it. Continuity of functions is one of the core concepts of
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are p ...
, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are
real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
s. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In
order theory Order theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematic ...
, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article. As an example, the function denoting the height of a growing flower at time would be considered continuous. In contrast, the function denoting the amount of money in a bank account at time would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.


History

A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817.
Augustin-Louis Cauchy
Augustin-Louis Cauchy
defined continuity of y=f(x) as follows: an infinitely small increment \alpha of the independent variable ''x'' always produces an infinitely small change f(x+\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 from
real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
s to real numbers, can be represented by a graph of a function, graph in the Cartesian coordinate system, Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain of a function, domain is the entire real line. A more mathematically rigorous definition is given below. A rigorous definition of continuity of real functions is usually given in a first course in calculus in terms of the idea of a limit (mathematics), limit. First, a function with variable is said to be continuous ''at the point'' on the real line, if the limit of , as approaches that point , is equal to the value ; and second, the ''function (as a whole)'' is said to be ''continuous'', if it is continuous at every point. A function is said to be ''discontinuous'' (or to have a ''discontinuity'') at some point when it is not continuous there. These points themselves are also addressed as ''discontinuities''. There are several different definitions of continuity of a function. Sometimes a function is said to be continuous if it is continuous at every point in its domain. In this case, the function , with the domain of all real ,  any integer, is continuous. Sometimes an exception is made for boundaries of the domain. For example, the graph of the function , with the domain of all non-negative reals, has a ''left-hand'' endpoint. In this case only the limit from the ''right'' is required to equal the value of the function. Under this definition ''f'' is continuous at the boundary and so for all non-negative arguments. The most common and restrictive definition is that a function is continuous if it is continuous at all real numbers. In this case, the previous two examples are not continuous, but every polynomial function is continuous, as are the sine, cosine, and exponential functions. Care should be exercised in using the word ''continuous'', so that it is clear from the context which meaning of the word is intended. Using mathematical notation, there are several ways to define continuous functions in each of the three senses mentioned above. Let :f\colon D \rightarrow \R \quad be a function defined on a subset D of the set \R of real numbers. This subset D is the domain of a function, domain of ''f''. Some possible choices include :D = \R \quad ( D is the whole set of real numbers), or, for ''a'' and ''b'' real numbers, :D = [a, b] = \ \quad ( D is a closed interval), or :D = (a, b) = \ \quad ( 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 ''f'' is ''continuous at some point'' ''c'' of its domain if the limit of a function, limit 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 :\lim_ = f(c). In detail this means three conditions: first, ''f'' has to be defined at ''c'' (guaranteed by the requirement that ''c'' is in the domain of ''f''). Second, the limit on the left hand side of that equation has to exist. Third, the value of this limit must equal ''f''(''c''). The formal definition of a limit implies that every function is continuous at every isolated point of its domain.


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''(''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)\in N_1(f(c)) whenever x\in N_2(c). This definition only requires that the domain and the codomain are topological spaces and is thus the most general definition. It follows from this definition that a function ''f'' is automatically continuous at every isolated point of its domain. As a specific example, every real valued function on the set of integers is continuous.


Definition in terms of limits of sequences

One can instead require that for any sequence (mathematics), sequence (x_n)_ of points in the domain which convergent sequence, converges to ''c'', the corresponding sequence \left(f(x_n)\right)_ converges to ''f''(''c''). 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'' → ''R'' 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 number ''ε'' > 0, however small, there exists some number ''δ'' > 0 such that for all ''x'' in the domain of ''f'' with ''x''0 − ''δ'' < ''x'' < ''x''0 + ''δ'', the value of ''f''(''x'') satisfies : f(x_0) - \varepsilon < f(x) < f(x_0) + \varepsilon. Alternatively written, continuity of ''f'' : ''D'' → ''R'' at ''x''0 ∈ ''D'' means that for every ''ε'' > 0 there exists a ''δ'' > 0 such that for all ''x'' ∈ ''D'' : :, x - x_0 , < \delta \Rightarrow , f(x) - f(x_0) , < \varepsilon. More intuitively, we can say that if we want to get all the ''f''(''x'') values to stay in some small topological neighbourhood, neighborhood around ''f''(''x''0), 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'') 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 (topology), basis for the topology, here the metric topology. Weierstrass had required that the interval ''x''0 − ''δ'' < ''x'' < ''x''0 + ''δ'' 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 formalise this to a definition of continuity. A function C: [0,\infty) \to [0,\infty] is called a control function if * ''C'' is non decreasing * \inf_ C(\delta) = 0 A function ''f'' : ''D'' → ''R'' is ''C''-continuous at ''x''0 if :: , f(x) - f(x_0), \le C(, x- x_0, ) for all x \in D A function is continuous in ''x''0 if it is ''C''-continuous for some control function ''C''. This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions \mathcal a function is \mathcal-continuous if it is C-continuous for some C \in \mathcal. For example, the Lipschitz continuity, Lipschitz and Hölder continuous functions of exponent α below are defined by the set of control functions ::\mathcal_ = \ respectively ::\mathcal_ = \ .


Definition using oscillation

Continuity can also be defined in terms of Oscillation (mathematics), oscillation: a function ''f'' is continuous at a point ''x''0 if and only if its oscillation at that point is zero; in symbols, \omega_f(x_0) = 0. A benefit of this definition is that it ''quantifies'' discontinuity: the oscillation gives how ''much'' 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 ''ε'' (hence a G-delta set, Gδ set) – and gives a very quick proof of one direction of the Lebesgue integrability condition. The oscillation is equivalent to the ''ε''-''δ'' 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 ''ε''0 there is no ''δ'' that satisfies the ''ε''-''δ'' definition, then the oscillation is at least ''ε''0, and conversely if for every ''ε'' there is a desired ''δ'', the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.


Definition using the hyperreals

Cauchy defined continuity of a function in the following intuitive terms: an infinitesimal 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. :A real-valued function ''f'' is continuous at ''x'' if its natural extension to the hyperreals has the property that for all infinitesimal ''dx'', is infinitesimal (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
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\colon D \to \R, then the ''sum of continuous functions'' :s = f + g (defined by s(x) = f(x) + g(x) for all x\in D) is continuous in D. The same holds for the ''product of continuous functions,'' :p = f \cdot g (defined by p(x) = f(x) \cdot g(x) for all x \in D) is continuous in D. Combining the above preservations of continuity and the continuity of constant functions and of the identity function I(x) = x one arrives at the continuity of all polynomial, polynomial functions such as : (pictured on the right). In the same way it can be shown that the ''reciprocal of a continuous function'' :r = 1/f (defined by r(x) = 1/f(x) for all x \in D such that f(x) \ne 0) is continuous in D\setminus \. This implies that, excluding the roots of g, the ''quotient of continuous functions'' :q = f/g (defined by q(x) = f(x)/g(x) for all x \in D, such that g(x) \ne 0) is also continuous on D\setminus \. For example, the function (pictured) :y(x) = \frac is defined for all real numbers and is continuous at every such point. Thus it is a continuous function. The question of continuity at does not arise, since is not in the domain of ''y''. There is no continuous function ''F'': R → R that agrees with ''y''(''x'') for all . Since the function sine is continuous on all reals, the sinc function ''G''(''x'') = sin(''x'')/''x'', is defined and continuous for all real ''x'' ≠ 0. However, unlike the previous example, ''G'' ''can'' be extended to a continuous function on ''all'' real numbers, by ''defining'' the value ''G''(0) to be 1, which is the limit of ''G''(''x''), when ''x'' approaches 0, i.e., : G(0) = \lim_\frac = 1. Thus, by setting : G(x) = \begin \frac x & \textx \ne 0\\ 1 & \textx = 0, \end the sinc-function becomes a continuous function on all real numbers. The term ''removable singularity'' 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\colon D_g \subseteq \R \to R_g \subseteq\R\quad\text\quad f\colon D_f \subseteq \R\to R_f\subseteq D_g, their composition, denoted as c = g \circ f \colon D_f \to \R, and defined by c(x) = g(f(x)), is continuous. This construction allows stating, for example, that : e^ is continuous for all x > 0.


Examples of discontinuous functions

An example of a discontinuous function is the Heaviside step function H, defined by :H(x) = \begin 1 & \text x \ge 0\\ 0 & \text x < 0 \end Pick for instance \varepsilon = 1/2. Then there is no around x = 0, i.e. no open interval (-\delta,\;\delta) with \delta > 0, that will force all the H(x) values to be within the of H(0), i.e. within (1/2,\;3/2). Intuitively we can think of this type of discontinuity as a sudden Jump discontinuity, jump in function values. Similarly, the sign function, 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 \ne 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 (mathematics), pathological, for example, Thomae's function, :f(x)=\begin 1 &\textx=0\\ \frac&\textx=\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 for the set of rational numbers, :D(x)=\begin 0&\textx\text (\in \mathbb \smallsetminus \mathbb)\\ 1&\textx\text (\in \mathbb) \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(x_0)\ne y_0. Then f(x)\ne 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 : , f(x)-f(x_0), <\frac\quad \text\quad , x-x_0, <\delta Suppose there is a point in the neighbourhood , x-x_0, <\delta for which f(x)=y_0; then we have the contradiction : , f(x_0)-y_0, <\frac.


Intermediate value theorem

The intermediate value theorem is an existence theorem, based on the real number property of Real number#Completeness, completeness, and states: : If the real-valued function ''f'' is continuous on the interval (mathematics), closed interval [''a'', ''b''] and ''k'' is some number between ''f''(''a'') and ''f''(''b''), then there is some number ''c'' in [''a'', ''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 [''a'', ''b''] and ''f''(''a'') and ''f''(''b'') differ in Sign (mathematics), sign, then, at some point ''c'' in [''a'', ''b''], ''f''(''c'') must equal 0 (number), zero.


Extreme value theorem

The extreme value theorem states that if a function ''f'' is defined on a closed interval [''a'',''b''] (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists ''c'' ∈ [''a'',''b''] with ''f''(''c'') ≥ ''f''(''x'') for all ''x'' ∈ [''a'',''b'']. The same is true of the minimum of ''f''. These statements are not, in general, true if the function is defined on an open interval (''a'',''b'') (or any set that is not both closed and bounded), as, for example, the continuous function ''f''(''x'') = 1/''x'', defined on the open interval (0,1), does not attain a maximum, being unbounded above.


Relation to differentiability and integrability

Every differentiable function :f\colon (a, b) \rightarrow \mathbf R is continuous, as can be shown. The Theorem#Converse, converse does not hold: for example, the absolute value function :f(x)=, x, = \begin \;\;\ x & \textx \geq 0\\ -x & \textx < 0 \end is everywhere continuous. However, it is not differentiable at ''x'' = 0 (but is so everywhere else). Weierstrass function, 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(). More generally, the set of functions :f\colon \Omega \rightarrow \mathbf R (from an open interval (or open subset of R) Ω 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''(Ω). 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#Smoothness of curves and surfaces, Smoothness of curves and surfaces. Every continuous function :f\colon [a, b] \rightarrow \mathbf R is integrable function, 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 a sequence (mathematics), sequence :f_1, f_2, \dotsc \colon I \rightarrow \mathbf R of functions such that the limit :f(x) := \lim_ f_n(x) exists for all ''x'' in ''D'', the resulting function ''f''(''x'') is referred to as the pointwise convergence, pointwise limit of the sequence of functions (''f''''n'')''n''∈N. 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 uniform convergence, converges uniformly, by the uniform convergence theorem. This theorem can be used to show that the exponential functions, logarithms, square root function, and trigonometric functions are continuous.


Directional and semi-continuity

Image:Right-continuous.svg, A right-continuous function Image:Left-continuous.svg, A left-continuous function
Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi-continuity. Roughly speaking, a function is ''right-continuous'' 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 ''ε'' > 0 however small, there exists some number ''δ'' > 0 such that for all ''x'' in the domain with , the value of ''f''(''x'') will satisfy : , f(x) - f(c), < \varepsilon. This is the same condition as for continuous functions, except that it is required to hold for ''x'' strictly larger than ''c'' only. Requiring it instead for all ''x'' with yields the notion of ''left-continuous'' functions. A function is continuous if and only if it is both right-continuous and left-continuous. A function ''f'' is ''Semi-continuity, lower semi-continuous'' if, roughly, any jumps that might occur only go down, but not up. That is, for any ''ε'' > 0, there exists some number ''δ'' > 0 such that for all ''x'' in the domain with , the value of ''f''(''x'') satisfies :f(x) \geq f(c) - \epsilon. The reverse condition is ''Semi-continuity, upper semi-continuity''.


Continuous functions between metric spaces

The concept of continuous real-valued functions can be generalized to functions between metric spaces. A metric space is a set ''X'' equipped with a function (called metric (mathematics), 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 \colon X \times X \rightarrow \mathbf R that satisfies a number of requirements, notably the triangle inequality. Given two metric spaces (''X'', d''X'') and (''Y'', d''Y'') and a function :f\colon X \rightarrow Y then ''f'' is continuous at the point ''c'' in ''X'' (with respect to the given metrics) if for any positive real number ε, there exists a positive real number δ such that all ''x'' in ''X'' satisfying d''X''(''x'', ''c'') < δ will also satisfy d''Y''(''f''(''x''), ''f''(''c'')) < ε. As in the case of real functions above, this is equivalent to the condition that for every sequence (''x''''n'') in ''X'' with limit lim ''x''''n'' = ''c'', we have lim ''f''(''x''''n'') = ''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 (''x''''n'') in ''X'' with limit ''c'', the sequence (''f''(''x''''n'')) 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 Gδ set, Gδ set – this follows from the ε-δ 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 :T\colon V \rightarrow W between normed vector spaces ''V'' and ''W'' (which are vector spaces equipped with a compatible norm (mathematics), norm, denoted , , ''x'', , ) is continuous if and only if it is Bounded linear operator, 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 δ depends on ε and ''c'' in the definition above. Intuitively, a function ''f'' as above is uniformly continuous if the δ does not depend on the point ''c''. More precisely, it is required that for every
real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...
''ε'' > 0 there exists ''δ'' > 0 such that for every ''c'', ''b'' ∈ ''X'' with ''d''''X''(''b'', ''c'') < ''δ'', we have that ''d''''Y''(''f''(''b''), ''f''(''c'')) < ''ε''. Thus, any uniformly continuous function is continuous. The converse does not hold in general, but holds when the domain space ''X'' is compact topological space, compact. Uniformly continuous maps can be defined in the more general situation of uniform spaces. A function is Hölder continuity, Hölder continuous with exponent α (a real number) if there is a constant ''K'' such that for all ''b'' and ''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 is referred to as Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant ''K'' such that the inequality :d_Y (f(b), f(c)) \leq K \cdot d_X (b, c) holds for any ''b'', ''c'' in ''X''. The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.


Continuous functions between topological spaces

Another, more abstract, notion of continuity is continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of metric spaces. A topological space is a set ''X'' together with a topology on ''X'', which is a set of 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 neighbourhood (mathematics), neighbourhoods of a given point. The elements of a topology are called open subsets of ''X'' (with respect to the topology). A function :f\colon X \rightarrow Y between two topological spaces ''X'' and ''Y'' is continuous if for every open set ''V'' ⊆ ''Y'', the Image (mathematics)#Inverse image, 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 ''TX''), but the continuity of ''f'' depends on the topologies used on ''X'' and ''Y''. This is equivalent to the condition that the Image (mathematics)#Inverse image, 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\colon X \rightarrow 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 space, T0, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.


The translation in the language of neighborhoods of the (ε, δ)-definition of limit, (ε, δ)-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 , 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\rightarrow 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 δ-ε 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 Convergent filter, converges to x in X, which is expressed by writing \mathcal \to x, then necessarily f(\mathcal) \to f(x) in Y. If \mathcal(x) denotes the neighborhood filter at x then f : X \to Y is continuous at x if and only if f(\mathcal(x)) \to f(x) in Y. Moreover, this happens if and only if the prefilter f(\mathcal(x)) is a filter base for the neighborhood filter of f(x) in Y.


Alternative definitions

Several Characterizations of the category of topological spaces, equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.


Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of limit points. In many instances, this is accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed family, indexed by a directed set, known as net (mathematics), 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'' → ''Y'' is sequentially continuous if whenever a sequence (''x''''n'') in ''X'' converges to a limit ''x'', the sequence (''f''(''x''''n'')) 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 Axiom of countable choice, countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if ''X'' is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve 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\colon A\subseteq\mathbb\to\mathbb is continuous at x_0 (in the sense of (ε, δ)-definition of limit#Continuity, \epsilon-\delta continuity). Let (x_n)_ be a sequence converging at x_0 (such a sequence always exists, e.g. x_n=x,\forall 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 \forall n>\nu_ :, x_n-x_0, <\delta_, since (x_n) converges at x_0; combining this with (*) we obtain :\forall\epsilon>0\,\exists\nu_>0:\forall n>\nu_\quad, f(x_n)-f(x_0), <\epsilon. Assume on the contrary that f is sequentially continuous and proceed by contradiction: suppose f is not continuous at x_0 :\exists\epsilon>0:\forall\delta_>0,\,\exists x_: 0<, x_-x_0, <\delta_\epsilon\implies , f(x_)-f(x_0), >\epsilon then we can take \delta_=1/n,\,\forall n>0 and call the corresponding point x_ =:x_n: in this way we have defined a sequence (x_n)_ such that :\forall n>0\quad , x_n-x_0, <\frac,\quad , f(x_n)-f(x_0), >\epsilon by construction x_n\to x_0 but f(x_n)\not\to f(x_0), which contradicts the hypothesis of sequentially continuity. ∎


Closure operator and interior operator definitions

In terms of the Interior (topology), 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 (topology), closure operator, f : X \to Y is continuous if and only if for every subset A \subseteq X, :f\left(\operatorname_X A\right) ~\subseteq~ \operatorname_Y (f(A)). That is to say, given any element x \in X that belongs to the closure of a subset A \subseteq X, f(x) necessarily belongs to the closure of f(A) in Y. If we declare that a point x is a subset A \subseteq X if x \in \operatorname_X A, then this terminology allows for a plain English description of continuity: f is continuous if and only if for every subset A \subseteq X, f maps points that are close to A to points that are close to f(A). Similarly, f is continuous at a fixed given point x \in X if and only if whenever x is close to a subset A \subseteq X, then f(x) is close to f(A). Instead of specifying topological spaces by their Open set, open subsets, any topology on X can Equivalence of categories, alternatively be determined by a Kuratowski closure operator, closure operator or by an interior operator. Specifically, the map that sends a subset A of a topological space X to its Closure (topology), topological closure \operatorname_X A satisfies the Kuratowski closure axioms and conversely, for any Kuratowski closure operator, 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_X A of A in (X, \tau). If the sets X and Y are each associated with closure operators (both denoted by \operatorname) then a map f : X \to Y is continuous if and only if f\left(\operatorname A\right) \subseteq \operatorname (f(A)) for every subset A \subseteq X. Similarly, the map that sends a subset A of X to its Interior (topology), topological interior \operatorname_X A defines an interior operator and 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_X A of A in (X, \tau). If the sets X and Y are each associated with interior operators (both denoted by \operatorname) then a map f : X \to Y is continuous if and only if f^\left(\operatorname B\right) \subseteq \operatorname\left(f^(B)\right) for every subset B \subseteq Y.


Filters and prefilters

Continuity can also be characterized in terms of Filters in topology, filters. A function f : X \to Y is continuous if and only if whenever a Filter (mathematics), filter \mathcal on X Convergent filter, converges in X to a point x \in X, then the prefilter f(\mathcal) converges in Y to f(x). This characterization remains true if the word "filter" is replaced by "prefilter."


Properties

If ''f'': ''X'' → ''Y'' and ''g'': ''Y'' → ''Z'' are continuous, then so is the composition ''g'' ∘ ''f'': ''X'' → ''Z''. If ''f'': ''X'' → ''Y'' is continuous and * ''X'' is Compact space, compact, then ''f''(''X'') is compact. * ''X'' is Connected space, connected, then ''f''(''X'') is connected. * ''X'' is path-connected, then ''f''(''X'') is path-connected. * ''X'' is Lindelöf space, Lindelöf, then ''f''(''X'') is Lindelöf. * ''X'' is separable space, separable, then ''f''(''X'') is separable. The possible topologies on a fixed set ''X'' are partial ordering, partially ordered: a topology τ1 is said to be comparison of topologies, coarser than another topology τ2 (notation: τ1 ⊆ τ2) if every open subset with respect to τ1 is also open with respect to τ2. Then, the identity function, identity map :idX: (''X'', τ2) → (''X'', τ1) is continuous if and only if τ1 ⊆ τ2 (see also comparison of topologies). More generally, a continuous function :(X, \tau_X) \rightarrow (Y, \tau_Y) stays continuous if the topology τ''Y'' is replaced by a Comparison of topologies, coarser topology and/or τ''X'' is replaced by a Comparison of topologies, finer topology.


Homeomorphisms

Symmetric to the concept of a continuous map is an open map, for which ''images'' 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''−1 need not be continuous. A bijective continuous function with continuous inverse function is called a ''homeomorphism''. If a continuous bijection has as its Domain of a function, domain a compact space and its codomain is Hausdorff space, Hausdorff, then it is a homeomorphism.


Defining topologies via continuous functions

Given a function :f \colon X \rightarrow 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''−1(''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 Comparison of topologies, 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 \rightarrow X into all topological spaces ''X''. Duality (mathematics), Dually, a similar idea can be applied to maps X \rightarrow S.


Related notions

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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematic ...
, an order-preserving function ''f'': ''X'' → ''Y'' between particular types of partially ordered sets ''X'' and ''Y'' is Scott continuity, continuous if for each directed set, directed subset ''A'' of ''X'', we have sup(''f''(''A'')) = ''f''(sup(''A'')). Here sup is the supremum with respect to the orderings in ''X'' and ''Y'', respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the Scott topology. In category theory, a functor :F\colon \mathcal C \rightarrow \mathcal D between two category (mathematics), categories is called ''continuous functor, continuous'', if it commutes with small limit (category theory), limits. That is to say, :\varprojlim_ F(C_i) \cong F \left(\varprojlim_ C_i \right) for any small (i.e., indexed by a set ''I'', as opposed to a class (mathematics), class) diagram (category theory), diagram of object (category theory), objects in \mathcal C. A ''continuity space'' is a generalization of metric spaces and posets, which uses the concept of quantales, and that can be used to unify the notions of metric spaces and Domain theory, domains.


See also

* Absolute continuity * Classification of discontinuities * Coarse function * Continuous function (set theory) * Continuous stochastic process * Dini continuity * Equicontinuity * Normal function * Open and closed maps * Piecewise * Symmetrically continuous function * Direction-preserving function - an analogue of a continuous function in discrete spaces.


References


Bibliography

* * {{Analysis-footer Continuous mappings, Calculus Types of functions