Continuous function

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In mathematics, 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.

Contents

Continuity of functions is one of the core concepts of topology, 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 numbers. 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, 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 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account at time t would be considered discontinuous, since it "jumps" at each point in time when money is deposited or withdrawn.

History

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

Definition

The function
f
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{\displaystyle f(x)={\tfrac {1}{x}}}
is continuous on the domain
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{\displaystyle \mathbb {R} \setminus \{0\}}
, but is not continuous over the domain
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because it is undefined at
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{\displaystyle x=0} Function-1 x.svg
The function is continuous on the domain , but is not continuous over the domain because it is undefined at

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. [7]

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. First, a function f with variable x is said to be continuous at the pointc on the real line, if the limit of f(x), as x approaches that point c, is equal to the value f(c); 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 f(x)=tan(x), with the domain of all real x ≠ (2n+1)π/2, n any integer, is continuous. Sometimes an exception is made for boundaries of the domain. For example, the graph of the function f(x)=x, 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 x = 0 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

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

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

( is the whole set of real numbers), or, for a and b real numbers,
( is a closed interval), or
( is an open interval).

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

Definition in terms of limits of functions

The function f is continuous at some pointc of its domain if the limit of f(x), as x approaches c through the domain of f, exists and is equal to f(c). [8] 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 on the left hand side of that equation has to exist. Third, the value of this limit must equal f(c).

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

Definition in terms of neighborhoods

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

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

Definition in terms of limits of sequences

The sequence exp(1/n) converges to exp(0) Continuity of the Exponential at 0.svg
The sequence exp(1/n) converges to exp(0)

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

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

Illustration of the e-d-definition: for e=0.5, c=2, the value d=0.5 satisfies the condition of the definition. Example of continuous function.svg
Illustration of the ε-δ-definition: for ε=0.5, c=2, the value δ=0.5 satisfies the condition of the definition.

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 x0 of the domain D, f is said to be continuous at the point x0 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 x0  δ < x < x0 + δ, the value of f(x) satisfies

Alternatively written, continuity of f : D  R at x0  D means that for every ε > 0 there exists a δ > 0 such that for all x  D :

More intuitively, we can say that if we want to get all the f(x) values to stay in some small neighborhood around f(x0), we simply need to choose a small enough neighborhood for the x values around x0. If we can do that no matter how small the f(x) neighborhood is, then f is continuous at x0.

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 x0  δ < x < x0 + 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 is called a control function if

  • C is non decreasing

A function f : D  R is C-continuous at x0 if

for all

A function is continuous in x0 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. Rapid Oscillation.svg
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 x0 if and only if its oscillation at that point is zero; [9] 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 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 Gδ set) – and gives a very quick proof of one direction of the Lebesgue integrability condition. [10]

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 ε0 there is no δ that satisfies the ε-δ definition, then the oscillation is at least ε0, 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, f(x+dx) − f(x) is infinitesimal [11]

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

Construction of continuous functions

The graph of a cubic function has no jumps or holes. The function is continuous. Brent method example.svg
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

f(x) = x3 + x2 − 5x + 3

(pictured on the right).

The graph of a continuous rational function. The function is not defined for x=-2. The vertical and horizontal lines are asymptotes. Homografia.svg
The graph of a continuous rational function. The function is not defined for x=−2. 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 x ≠ −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: RR that agrees with y(x) for all x ≠ −2.

The sinc and the cos functions Si cos.svg
The sinc and the cos functions

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, Gcan 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.,

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
lim
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{\displaystyle \lim _{n\to \infty }\operatorname {sgn} \left({\tfrac {1}{n}}\right)\neq \operatorname {sgn} \left(\lim _{n\to \infty }{\tfrac {1}{n}}\right)}
. Thus, the signum function is discontinuous at 0 (see section 2.1.3). Discontinuity of the sign function at 0.svg
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. Thomae function (0,1).svg
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.

Properties

A useful lemma

Let be a function that is continuous at a point and be a value such Then throughout some neighbourhood of [12]

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 [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, then, at some point c in [a, b], f(c) must equal 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

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 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 C1((a, b)). More generally, the set of functions

(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 Cn(Ω). See differentiability class. In the field of computer graphics, properties related (but not identical) to C0, C1, C2 are sometimes called G0 (continuity of position), G1 (continuity of tangency), and G2 (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 fn(x) whose (pointwise) limit function f(x) is discontinuous. The convergence is not uniform. Uniform continuity animation.gif
A sequence of continuous functions fn(x) whose (pointwise) limit function f(x) is discontinuous. The convergence is not uniform.

Given a sequence

of functions such that the limit

exists for all x in D, the resulting function f(x) is referred to as the pointwise limit of the sequence of functions (fn)nN. The pointwise limit function need not be continuous, even if all functions fn are continuous, as the animation at the right shows. However, f is continuous if all functions fn 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 ε > 0 however small, there exists some number δ > 0 such that for all x in the domain with c<x<c + δ, the value of f(x) will satisfy

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 cδ<x<c 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 ε > 0, there exists some number δ > 0 such that for all x in the domain with |x − c| < δ, the value of f(x) 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 X equipped with a function (called metric) dX, 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 (X, dX) and (Y, dY) and a function

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 dX(x, c) < δ will also satisfy dY(f(x), f(c)) < ε. As in the case of real functions above, this is equivalent to the condition that for every sequence (xn) in X with limit lim xn = c, we have lim f(xn) = 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 (xn) in X with limit c, the sequence (f(xn)) 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  – 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

between normed vector spaces V and W (which are vector spaces equipped with a compatible norm, denoted ||x||) is continuous if and only if it is bounded, that is, there is a constant K such that

for all x in V.

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. Lipschitz continuity.png
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 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 ε > 0 there exists δ > 0 such that for every c, b  X with dX(b, c) < δ, we have that dY(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. Uniformly continuous maps can be defined in the more general situation of uniform spaces. [13]

A function is 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

holds. Any Hölder continuous function is uniformly continuous. The particular case α = 1 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 b, c in X. [14] 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 neighbourhoods 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 VY, 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 TX), 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 range is indiscrete is continuous.

Continuity at a point

Continuity at a point: For every neighborhood V of f(x), there is a neighborhood U of x such that f(U) [?] V Continuity topology.svg
Continuity at a point: For every neighborhood V of f(x), there is a neighborhood U of x such that f(U) ⊆ V

The translation in the language of neighborhoods of the (ε, δ)-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 x such that f(U) ⊆ V.

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 f(U) ⊆ V, this definition may be simplified into:

A function is continuous at a point if and only if is a neighborhood of x 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 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 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 [15] Moreover, this happens if and only if the prefilter is a filter base for the neighborhood filter of in [15]

Alternative definitions

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.

Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of limit points. In many instances, this is accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, 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: XY is sequentially continuous if whenever a sequence (xn) in X converges to a limit x, the sequence (f(xn)) converges to f(x). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If X is a first-countable space and countable choice 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: [16]

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

Proof

Proof. Assume that is continuous at (in the sense of continuity). Let be a sequence converging at (such a sequence always exists, e.g. ); since is continuous at

For any such we can find a natural number such that

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 and 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 and 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 [17]

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." [15]

Properties

If f: XY and g: YZ are continuous, then so is the composition gf: XZ. If f: XY is continuous and

The possible topologies on a fixed set X are partially ordered: a topology τ1 is said to be 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 map

idX: (X, τ2) → (X, τ1)

is continuous if and only if τ1 ⊆ τ2 (see also comparison of topologies). More generally, a continuous function

stays continuous if the topology τY is replaced by a coarser topology and/or τX is replaced by a 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 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 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 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 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 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 an 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. Were 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.

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 X and Y is continuous if for each directed subset A of X, we have Here 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. [18] [19]

In category theory, a functor

between two categories is called continuous , if it commutes with small limits. That is to say,

for any small (i.e., indexed by a set I, as opposed to a class) diagram of objects in .

A continuity space is a generalization of metric spaces and posets, [20] [21] which uses the concept of quantales, and that can be used to unify the notions of metric spaces and domains. [22]

See also

Related Research Articles

Intermediate value theorem A continuous function on an interval takes on every value between its values at the ends

In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f(a) and f(b) at some point within the interval.

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

Uniform continuity Uniform restraint of the change in functions

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y be sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.

Open set Basic subset of a topological space

In mathematics, open sets are a generalization of open intervals in the real line. In a metric space—that is, when a distance is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis. A topological vector space is a vector space which is also a topological space, this implies that vector space operations be continuous functions. More specifically, its topological space has a uniform topological structure, allowing a notion of uniform convergence.

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is closed under the limit operation. This should not be confused with a closed manifold.

Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

General topology Branch of topology

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.

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 operations of calculus—differentiation and integration. This relationship is commonly characterized in the framework of Riemann integration, but with absolute continuity it may be formulated in terms of Lebesgue integration. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These two notions are generalized in different directions. The usual derivative of a function is related to the Radon–Nikodym derivative, or density, of a measure.

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 is open if for any open set in the image is open in Likewise, a closed map is a function that maps closed sets to closed sets. A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

In functional and convex analysis, and related disciplines of mathematics, the polar set is a special convex set associated to any subset of a vector space lying in the dual space The bipolar of a subset is the polar of but lies in .

In general topology and related areas of mathematics, the final topology on a set with respect to a family of functions from topological spaces into is the finest topology on that makes all those functions continuous.

In topology and related fields of mathematics, a sequential space is a topological space that satisfies a very weak axiom of countability.

In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.

Filters in topology Use of filters to describe and characterize all basic topological notions and results.

In topology, a subfield of mathematics, filters are special families of subsets of a set that can be used to study topological spaces and define all basic topological notions such a convergence, continuity, compactness, and more. Filters also provide a common framework for defining various types of limits of functions such as limits from the left/right, to infinity, to a point or a set, and many others. Special types of filters called ultrafilters have many useful technical properties and they may often be used in place of arbitrary filters.

In functional analysis and related areas of mathematics, a complete topological vector space is a topological vector space (TVS) with the property that whenever points get progressively closer to each other, then there exists some point towards which they all get closer to. The notion of "points that get progressively closer" is made rigorous by Cauchy nets or Cauchy filters, which are generalizations of Cauchy sequences, while "point towards which they all get closer to" means that this net or filter converges to Unlike the notion of completeness for metric spaces, which it generalizes, the notion of completeness for TVSs does not depend on any metric and is defined for all TVSs, including those that are not metrizable or Hausdorff.

In mathematics, the injective tensor product of two topological vector spaces (TVSs) was introduced by Alexander Grothendieck and was used by him to define nuclear spaces. An injective tensor product is in general not necessarily complete, so its completion is called the completed injective tensor products. Injective tensor products have applications outside of nuclear spaces. In particular, as described below, up to TVS-isomorphism, many TVSs that are defined for real or complex valued functions, for instance, the Schwartz space or the space of continuously differentiable functions, can be immediately extended to functions valued in a Hausdorff locally convex TVS Y without any need to extend definitions from real/complex-valued functions to Y-valued functions.

In the mathematical discipline of functional analysis, it is possible to generalize the notion of derivative to arbitrary topological vector spaces (TVSs) in multiple ways. But when the domain of a TVS-value function is a subset of finite-dimensional Euclidean space then the number of generalizations of the derivative is much more limited and derivatives are more well behaved. This article presents the theory of -times continuously differentiable functions on an open subset of Euclidean space , which is an important special case of differentiation between arbitrary TVSs. All vector spaces will be assumed to be over the field where is either the real numbers or the complex numbers

In mathematics, specifically topology, a sequence covering map is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include sequentially quotient maps, sequence coverings, 1-sequence coverings, and 2-sequence coverings. These classes of maps are closely related to sequential spaces. If the domain and/or codomain have certain additional topological properties then these definitions become equivalent to other well-known classes of maps, such as open maps or quotient maps, for example. In these situations, characterizations of such properties in terms of convergent sequences might provide benefits similar to those provided by, say for instance, the characterization of continuity in terms of sequential continuity or the characterization of compactness in terms of sequential compactness.

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