In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous. Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.

Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of topology.

A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity.

As an example, the function H(t) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M(t) denoting the amount of money in a bank account at time t would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.



A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of   as follows: an infinitely small increment   of the independent variable x always produces an infinitely small change   of the dependent variable y (see e.g. Cours d'Analyse, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity). The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano,[1] Karl Weierstrass[2] denied continuity of a function at a point c unless it was defined at and on both sides of c, but Édouard Goursat[3] allowed the function to be defined only at and on one side of c, and Camille Jordan[4] allowed it even if the function was defined only at c. All three of those nonequivalent definitions of pointwise continuity are still in use.[5] 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.[6]

Real functions



The function   is continuous on its domain ( ), but is discontinuous at   when considered as a partial function defined on the reals.[7].

A real function that is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition is given below.[8]

Continuity of real functions is usually defined in terms of limits. A function f with variable x is continuous at the real number c, if the limit of   as x tends to c, is equal to  

There are several different definitions of the (global) continuity of a function, which depend on the nature of its domain.

A function is continuous on an open interval if the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval   (the whole real line) is often called simply a continuous function; one also says that such a function is continuous everywhere. For example, all polynomial functions are continuous everywhere.

A function is continuous on a semi-open or a closed interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function   is continuous on its whole domain, which is the closed interval  

Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some isolated points. Examples include the reciprocal function   and the tangent function   When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous.

A partial function is discontinuous at a point if the point belongs to the topological closure of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions   and   are discontinuous at 0, and remain discontinuous whichever value is chosen for defining them at 0. A point where a function is discontinuous is called a discontinuity.

Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above.

Let   be a function defined on a subset   of the set   of real numbers.

This subset   is the domain of f. Some possible choices include

  •  : i.e.,   is the whole set of real numbers. or, for a and b real numbers,
  •  :   is a closed interval, or
  •  :   is an open interval.

In the case of the domain   being defined as an open interval,   and   do not belong to  , and the values of   and   do not matter for continuity on  .

Definition in terms of limits of functions


The function f is continuous at some point c of its domain if the limit of   as x approaches c through the domain of f, exists and is equal to  [9] In mathematical notation, this is written as   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 of that equation has to exist. Third, the value of this limit must equal  

(Here, we have assumed that the domain of f does not have any isolated points.)

Definition in terms of neighborhoods


A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point   as the width of the neighborhood around c shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood   there is a neighborhood   in its domain such that   whenever  

As neighborhoods are defined in any topological space, this definition of a continuous function applies not only for real functions but also when the domain and the codomain are topological spaces and is thus the most general definition. It follows that a function is automatically continuous at every isolated point of its domain. For example, every real-valued function on the integers is continuous.

Definition in terms of limits of sequences

The sequence exp(1/n) converges to exp(0) = 1

One can instead require that for any sequence   of points in the domain which converges to c, the corresponding sequence   converges to   In mathematical notation,  

Weierstrass and Jordan definitions (epsilon–delta) of continuous functions

Illustration of the ε-δ-definition: at x = 2, any value δ ≤ 0.5 satisfies the condition of the definition for ε = 0.5.

Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function   as above and an element   of the domain  ,   is said to be continuous at the point   when the following holds: For any positive real number   however small, there exists some positive real number   such that for all   in the domain of   with   the value of   satisfies  

Alternatively written, continuity of   at   means that for every   there exists a   such that for all  :  

More intuitively, we can say that if we want to get all the   values to stay in some small neighborhood around   we need to choose a small enough neighborhood for the   values around   If we can do that no matter how small the   neighborhood is, then   is continuous at  

In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology, here the metric topology.

Weierstrass had required that the interval   be entirely within the domain  , but Jordan removed that restriction.

Definition in terms of control of the remainder


In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity. A function   is called a control function if

  • C is non-decreasing

A function   is C-continuous at   if there exists such a neighbourhood   that  

A function is continuous in   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   a function is  -continuous if it is  -continuous for some   For example, the Lipschitz and Hölder continuous functions of exponent α below are defined by the set of control functions   respectively  

Definition using oscillation

The failure of a function to be continuous at a point is quantified by its oscillation.

Continuity can also be defined in terms of oscillation: a function f is continuous at a point   if and only if its oscillation at that point is zero;[10] in symbols,   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 helpful 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   (hence a   set) – and gives a rapid proof of one direction of the Lebesgue integrability condition.[11]

The oscillation is equivalent to the   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   there is no   that satisfies the   definition, then the oscillation is at least   and conversely if for every   there is a desired   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 the continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34). Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows.

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[12]

(see microcontinuity). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy's definition of continuity.

Construction of continuous functions

The graph of a cubic function has no jumps or holes. The function is continuous.

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   then the sum of continuous functions   (defined by   for all  ) is continuous in  

The same holds for the product of continuous functions,   (defined by   for all  ) is continuous in  

Combining the above preservations of continuity and the continuity of constant functions and of the identity function   on  , one arrives at the continuity of all polynomial functions on  , such as   (pictured on the right).

The graph of a continuous rational function. The function is not defined for   The vertical and horizontal lines are asymptotes.

In the same way, it can be shown that the reciprocal of a continuous function   (defined by   for all   such that  ) is continuous in  

This implies that, excluding the roots of   the quotient of continuous functions   (defined by   for all  , such that  ) is also continuous on  .

For example, the function (pictured)   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   There is no continuous function   that agrees with   for all  

The sinc and the cos functions

Since the function sine is continuous on all reals, the sinc function   is defined and continuous for all real   However, unlike the previous example, G can be extended to a continuous function on all real numbers, by defining the value   to be 1, which is the limit of   when x approaches 0, i.e.,  

Thus, by setting


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   their composition, denoted as   and defined by   is continuous.

This construction allows stating, for example, that   is continuous for all  

Examples of discontinuous functions

Plot of the signum function. It shows that  . Thus, the signum function is discontinuous at 0 (see section 2.1.3).

An example of a discontinuous function is the Heaviside step function  , defined by  

Pick for instance  . Then there is no  -neighborhood around  , i.e. no open interval   with   that will force all the   values to be within the  -neighborhood of  , i.e. within  . Intuitively, we can think of this type of discontinuity as a sudden jump in function values.

Similarly, the signum or sign function   is discontinuous at   but continuous everywhere else. Yet another example: the function   is continuous everywhere apart from  .

Point plot of Thomae's function on the interval (0,1). The topmost point in the middle shows f(1/2) = 1/2.

Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined pathological, for example, Thomae's function,   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,   is nowhere continuous.



A useful lemma


Let   be a function that is continuous at a point   and   be a value such   Then   throughout some neighbourhood of  [13]

Proof: By the definition of continuity, take   , then there exists   such that   Suppose there is a point in the neighbourhood   for which   then we have the contradiction  

Intermediate value theorem


The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states:

If the real-valued function f is continuous on the closed interval   and k is some number between   and   then there is some number   such that  

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   and   and   differ in sign, then, at some point     must equal zero.

Extreme value theorem


The extreme value theorem states that if a function f is defined on a closed interval   (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists   with   for all   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   (or any set that is not both closed and bounded), as, for example, the continuous function   defined on the open interval (0,1), does not attain a maximum, being unbounded above.

Relation to differentiability and integrability


Every differentiable function   is continuous, as can be shown. The converse does not hold: for example, the absolute value function


is everywhere continuous. However, it is not differentiable at   (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   More generally, the set of functions   (from an open interval (or open subset of  )   to the reals) such that f is   times differentiable and such that the  -th derivative of f is continuous is denoted   See differentiability class. In the field of computer graphics, properties related (but not identical) to   are sometimes called   (continuity of position),   (continuity of tangency), and   (continuity of curvature); see Smoothness of curves and surfaces.

Every continuous function   is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the (integrable but discontinuous) sign function shows.

Pointwise and uniform limits

A sequence of continuous functions   whose (pointwise) limit function   is discontinuous. The convergence is not uniform.

Given a sequence   of functions such that the limit   exists for all  , the resulting function   is referred to as the pointwise limit of the sequence of functions   The pointwise limit function need not be continuous, even if all functions   are continuous, as the animation at the right shows. However, f is continuous if all functions   are continuous and the sequence 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


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   however small, there exists some number   such that for all x in the domain with   the value of   will satisfy  

This is the same condition as continuous functions, except it is required to hold for x strictly larger than c only. Requiring it instead for all x with   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 lower semi-continuous if, roughly, any jumps that might occur only go down, but not up. That is, for any   there exists some number   such that for all x in the domain with   the value of   satisfies   The reverse condition is 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   equipped with a function (called metric)   that can be thought of as a measurement of the distance of any two elements in X. Formally, the metric is a function   that satisfies a number of requirements, notably the triangle inequality. Given two metric spaces   and   and a function   then   is continuous at the point   (with respect to the given metrics) if for any positive real number   there exists a positive real number   such that all   satisfying   will also satisfy   As in the case of real functions above, this is equivalent to the condition that for every sequence   in   with limit   we have   The latter condition can be weakened as follows:   is continuous at the point   if and only if for every convergent sequence   in   with limit  , the sequence   is a Cauchy sequence, and   is in the domain of  .

The set of points at which a function between metric spaces is continuous is a   set – this follows from the   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   between normed vector spaces   and   (which are vector spaces equipped with a compatible norm, denoted  ) is continuous if and only if it is bounded, that is, there is a constant   such that   for all  

Uniform, Hölder and Lipschitz continuity

For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph so that the graph always remains entirely outside the cone.

The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way   depends on   and c in the definition above. Intuitively, a function f as above is uniformly continuous if the   does not depend on the point c. More precisely, it is required that for every real number   there exists   such that for every   with   we have that   Thus, any uniformly continuous function is continuous. The converse does not generally hold but holds when the domain space X is compact. Uniformly continuous maps can be defined in the more general situation of uniform spaces.[14]

A function is Hölder continuous with exponent α (a real number) if there is a constant K such that for all   the inequality   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   holds for any  [15] 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 the 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 one to talk about the neighborhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology).

A function   between two topological spaces X and Y is continuous if for every open set   the inverse image   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  ), 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   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 codomain is indiscrete is continuous.

Continuity at a point

Continuity at a point: For every neighborhood V of  , there is a neighborhood U of x such that  

The translation in the language of neighborhoods of the  -definition of continuity leads to the following definition of the continuity at a point:

A function   is continuous at a point   if and only if for any neighborhood V of   in Y, there is a neighborhood U of   such that  

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   is the largest subset U of X such that   this definition may be simplified into:

A function   is continuous at a point   if and only if   is a neighborhood of   for every neighborhood V of   in Y.

As an open set is a set that is a neighborhood of all its points, a function   is continuous at every point of X 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   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   a map   is continuous at   if and only if whenever   is a filter on   that converges to   in   which is expressed by writing   then necessarily   in   If   denotes the neighborhood filter at   then   is continuous at   if and only if   in  [16] Moreover, this happens if and only if the prefilter   is a filter base for the neighborhood filter of   in  [16]

Alternative definitions


Several equivalent definitions for a topological structure exist; thus, several equivalent ways exist to define a continuous function.

Sequences and nets


In several contexts, the topology of a space is conveniently specified in terms of limit points. This is often accomplished by specifying when a point is the limit of a sequence. Still, for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, 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   is sequentially continuous if whenever a sequence   in   converges to a limit   the sequence   converges to   Thus, sequentially continuous functions "preserve sequential limits." Every continuous function is sequentially continuous. If   is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if   is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve the limits of nets, and this property characterizes continuous functions.

For instance, consider the case of real-valued functions of one real variable:[17]

Theorem — A function   is continuous at   if and only if it is sequentially continuous at that point.


Proof. Assume that   is continuous at   (in the sense of   continuity). Let   be a sequence converging at   (such a sequence always exists, for example,  ); since   is continuous at     For any such   we can find a natural number   such that for all     since   converges at  ; combining this with   we obtain   Assume on the contrary that   is sequentially continuous and proceed by contradiction: suppose   is not continuous at     then we can take   and call the corresponding point  : in this way we have defined a sequence   such that   by construction   but  , which contradicts the hypothesis of sequentially continuity.  

Closure operator and interior operator definitions


In terms of the interior operator, a function   between topological spaces is continuous if and only if for every subset    

In terms of the closure operator,   is continuous if and only if for every subset     That is to say, given any element   that belongs to the closure of a subset     necessarily belongs to the closure of   in   If we declare that a point   is close to a subset   if   then this terminology allows for a plain English description of continuity:   is continuous if and only if for every subset     maps points that are close to   to points that are close to   Similarly,   is continuous at a fixed given point   if and only if whenever   is close to a subset   then   is close to  

Instead of specifying topological spaces by their open subsets, any topology on   can alternatively be determined by a closure operator or by an interior operator. Specifically, the map that sends a subset   of a topological space   to its topological closure   satisfies the Kuratowski closure axioms. Conversely, for any closure operator   there exists a unique topology   on   (specifically,  ) such that for every subset     is equal to the topological closure   of   in   If the sets   and   are each associated with closure operators (both denoted by  ) then a map   is continuous if and only if   for every subset  

Similarly, the map that sends a subset   of   to its topological interior   defines an interior operator. Conversely, any interior operator   induces a unique topology   on   (specifically,  ) such that for every     is equal to the topological interior   of   in   If the sets   and   are each associated with interior operators (both denoted by  ) then a map   is continuous if and only if   for every subset  [18]

Filters and prefilters


Continuity can also be characterized in terms of filters. A function   is continuous if and only if whenever a filter   on   converges in   to a point   then the prefilter   converges in   to   This characterization remains true if the word "filter" is replaced by "prefilter."[16]



If   and   are continuous, then so is the composition   If   is continuous and

The possible topologies on a fixed set X are partially ordered: a topology   is said to be coarser than another topology   (notation:  ) if every open subset with respect to   is also open with respect to   Then, the identity map   is continuous if and only if   (see also comparison of topologies). More generally, a continuous function   stays continuous if the topology   is replaced by a coarser topology and/or   is replaced by a finer topology.



Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. 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   need not be continuous. A bijective continuous function with a continuous inverse function is called a homeomorphism.

If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.

Defining topologies via continuous functions


Given a function   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   is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on S. Thus, the final topology is the finest topology on S that makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f.

Dually, for a function f from a set S to a topological space X, the initial topology on S is defined by designating as an open set every subset A of S such that   for some open subset U of X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus, the initial topology is the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X.

A topology on a set S is uniquely determined by the class of all continuous functions   into all topological spaces X. Dually, a similar idea can be applied to maps  


If   is a continuous function from some subset   of a topological space   then a continuous extension of   to   is any continuous function   such that   for every   which is a condition that often written as   In words, it is any continuous function   that restricts to   on   This notion is used, for example, in the Tietze extension theorem and the Hahn–Banach theorem. If   is not continuous, then it could not possibly have a continuous extension. If   is a Hausdorff space and   is a dense subset of   then a continuous extension of   to   if one exists, will be unique. The Blumberg theorem states that if   is an arbitrary function then there exists a dense subset   of   such that the restriction   is continuous; in other words, every function   can be restricted to some dense subset on which it is continuous.

Various other mathematical domains use the concept of continuity in different but related meanings. For example, in order theory, an order-preserving function   between particular types of partially ordered sets   and   is continuous if for each directed subset   of   we have   Here   is the supremum with respect to the orderings in   and   respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the Scott topology.[19][20]

In category theory, a functor   between two categories is called continuous if it commutes with small limits. That is to say,   for any small (that is, indexed by a set   as opposed to a class) diagram of objects in  .

A continuity space is a generalization of metric spaces and posets,[21][22] which uses the concept of quantales, and that can be used to unify the notions of metric spaces and domains.[23]

See also



  1. ^ Bolzano, Bernard (1817). "Rein analytischer Beweis des Lehrsatzes daß zwischen je zwey Werthen, die ein entgegengesetzetes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege". Prague: Haase.
  2. ^ Dugac, Pierre (1973), "Eléments d'Analyse de Karl Weierstrass", Archive for History of Exact Sciences, 10 (1–2): 41–176, doi:10.1007/bf00343406, S2CID 122843140
  3. ^ Goursat, E. (1904), A course in mathematical analysis, Boston: Ginn, p. 2
  4. ^ Jordan, M.C. (1893), Cours d'analyse de l'École polytechnique, vol. 1 (2nd ed.), Paris: Gauthier-Villars, p. 46
  5. ^ Harper, J.F. (2016), "Defining continuity of real functions of real variables", BSHM Bulletin: Journal of the British Society for the History of Mathematics, 31 (3): 1–16, doi:10.1080/17498430.2015.1116053, S2CID 123997123
  6. ^ Rusnock, P.; Kerr-Lawson, A. (2005), "Bolzano and uniform continuity", Historia Mathematica, 32 (3): 303–311, doi:10.1016/
  7. ^ Strang, Gilbert (1991). Calculus. SIAM. p. 702. ISBN 0961408820.
  8. ^ Speck, Jared (2014). "Continuity and Discontinuity" (PDF). MIT Math. p. 3. Archived from the original (PDF) on 2016-10-06. Retrieved 2016-09-02. Example 5. The function   is continuous on   and on  , i.e., for   and for   in other words, at every point in its domain. However, it is not a continuous function since its domain is not an interval. It has a single point of discontinuity, namely  , and an infinite discontinuity there.
  9. ^ Lang, Serge (1997), Undergraduate analysis, Undergraduate Texts in Mathematics (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94841-6, section II.4
  10. ^ Introduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172
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