continuous function

Continuous functionTopics in Calculus

Fundamental theorem Limits of functions

Continuity Mean value theorem

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This article may require improve this article if you can. The

In mathematics, a continuous functioninput result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous

Continuity of functions is one of the core concepts of below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbersmore general case of functions between two theory, one considers a notion of continuity known as do exist but they are not discussed in this article.

Continuous function

may require cleanup to meet Wikipedia's quality standardsif you can. The talk page may contain suggestions.

continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A

function with a continuous inverse function is called "bicontinuous

e 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

real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in

, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.

Continuous function

quality standards. Please may contain suggestions. (June 2010)

for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A

bicontinuous".

, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and

n, this article discusses the definition for the , especially in domain

. Other forms of continuity

As an example, consider the function h(t) which describes the height of a growing flower at time t. This function is continuous. In fact, there is a dictum of classical physics which states that in nature everything is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous. (However, if one assumes a discrete set as the domain of function M, for instance the set of points of time at 4:00 PM on business days, then M becomes continuous function, as every function whose domain is a discrete subset of reals is.)

Contents

[hide]

• 1 Real-valued continuous functions o 1.1 Historical infinitesimal definition o 1.2 Definition in terms of limits o 1.3 Weierstrass definition (epsilon-delta) of continuous functions o 1.4 Definition using oscillation o 1.5 Definition using the hyperreals o 1.6 Examples o 1.7 Facts about continuous functions

1.7.1 Intermediate value theorem 1.7.2 Extreme value theorem

o 1.8 Directional continuity • 2 Continuous functions between metric spaces • 3 Continuous functions between topological spaces

o 3.1 Definitions 3.1.1 Open and closed set definition 3.1.2 Neighborhood definition 3.1.3 Sequences and nets 3.1.4 Closure operator definition 3.1.5 Closeness relation definition

o 3.2 Useful properties of continuous maps o 3.3 Other notes

• 4 Continuous functions between partially ordered sets • 5 Continuous binary relation • 6 Continuity space • 7 See also • 8 Notes • 9 References

[edit] Real-valued continuous functions

[edit] Historical infinitesimal definition

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).

[edit] Definition in terms of limits

Suppose we have a function that maps real numbers to real numbers and whose domain is some interval, like the functions h and M above. Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps".

In general, we say that the function f is continuous at some point c of its domain if, and only if, the following holds:

• The limit of f(x) as x approaches c through domain of f does exist and is equal to f(c); in

mathematical notation, . If the point c in the domain of f is not a limit point of the domain, then this condition is vacuously true, since x cannot approach c through values not equal c. Thus, for example, every function whose domain is the set of all integers is continuous.

We call a function continuous if and only if it is continuous at every point of its domain. More generally, we say that a function is continuous on some subset of its domain if it is continuous at every point of that subset.

The notation C(Ω) or C0(Ω) is sometimes used to denote the set of all continuous functions with domain Ω. Similarly, C1(Ω) is used to denote the set of differentiable functions whose derivative is continuous, C²(Ω) for the twice-differentiable functions whose second derivative is continuous, and so on (see differentiability class). In the field of computer graphics, these three levels are sometimes called g0 (continuity of position), g1 (continuity of tangency), and g2 (continuity of curvature). The notation C(n, α)(Ω) occurs in the definition of a more subtle concept, that of Hölder continuity.

[edit] Weierstrass definition (epsilon-delta) of continuous functions

Without resorting to limits, one can define continuity of real functions as follows.

Again consider a function ƒ that maps a set of real numbers to another set of real numbers, and suppose c is an element of the domain of ƒ. The function ƒ is said to be 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 of ƒ with c − δ < x < c + δ, the value of ƒ(x) satisfies

Alternatively written: Given subsets I, D of R, continuity of ƒ : I → D at c ∈ I means that for every ε > 0 there exists a δ > 0 such that for all x ∈ I,:

A form of this epsilon-delta definitionPreliminary forms of a related definition of the limit were given by definition and the distinction between pointwise continuity and given by Karl Weierstrass.

More intuitively, we can say that if we want to get all the neighborhood around ƒ(c), we simply need to choose a small enough neighborhood for the values around c, and we can do that no matter how small the continuous at c.

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

[edit] Definition using oscillation

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

Continuity can also be defined in terms of and only if the oscillation is zero;quantifies discontinuity: the oscillation gives how

This definition is useful in descriptive set theorycontinuous 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 integrability condition.[3]

The oscillation is equivalence to the limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given satisfies the ε-δ definition, thea desired δ, the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.

delta definition of continuity was first given by Bernard Preliminary forms of a related definition of the limit were given by Cauchydefinition and the distinction between pointwise continuity and uniform continuity

More intuitively, we can say that if we want to get all the ƒ(x) values to stay in some small ), we simply need to choose a small enough neighborhood for the

, and we can do that no matter how small the ƒ(x) neighborhood is;

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

] 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 ƒ is continuous at a point and only if the oscillation is zero;[2] in symbols, ωf(x0) = 0. A benefit of this definition is that it

discontinuity: the oscillation gives how much the function is discontinuous at a point.

descriptive set theory to study the set of discontinuities and the continuous points are the intersection of the sets where the oscillation is

and gives a very quick proof of one direction of the

The oscillation is equivalence to the ε-δ definition by a simple re-arrangement, and by using a ) to define oscillation: if (at a given point) for a given

definition, then the oscillation is at least ε0, and conversely if for every the oscillation is 0. The oscillation definition can be naturally generalized to maps

from a topological space to a metric space.

Bernard Bolzano in 1817. Cauchy,[1] though the formal

uniform continuity were first

) values to stay in some small ), we simply need to choose a small enough neighborhood for the x

) neighborhood is; ƒ is then

In modern terms, this is generalized by the definition of continuity of a function with respect to a

oscillation.

: a function ƒ is continuous at a point x0 if A benefit of this definition is that it

the function is discontinuous at a point.

to study the set of discontinuities and the continuous points are the intersection of the sets where the oscillation is

and gives a very quick proof of one direction of the Lebesgue

arrangement, and by using a ) to define oscillation: if (at a given point) for a given ε0 there is no δ that

, and conversely if for every ε there is the oscillation is 0. The oscillation definition can be naturally generalized to maps

[edit] Definition using the hyperreals

Non-standard analysis is a way of making 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 function ƒ from the reals to the reals is continuous if its natural extension to the hyperreals has the property that for real infinitesimal.[4]

In other words, an infinitesimal increment of the independent variable corresponds to an infinitesimal change of the dependent variable, giving a modern expression to Cauchy's definition of continuity.

[edit] Examples

• All polynomial functions• If a function has a domain which is not an interval, the notion of a continuous function as

one whose graph you can draw without taking your pencil off the paper is not quite correct. Consider the functionsat x = 0, so each has domain continuous. The question of continuity at domain of f nor in the domain of function whose domain is function will not be continuous. On the other hand, since the limit of extended continuously to

• The exponential functionsabsolute value function are continuous. continuous on all of R

• An example of a rational continuous function is 2 does not arise, since

• An example of a discontinuous function is the function = 0 if x ≤ 0. Pick for instance force all the f(x) values to be within discontinuity as a sudden jump in function values.

• Another example of a discontinuous function is the • A more complicated example of a discontinuous function is • Dirichlet's function

is nowhere continuous.

[edit] Facts about continuous functions

] Definition using the hyperreals

is a way of making Newton-Leibniz-style infinitesimalsrigorous. The real line is augmented by the addition of infinite and infinitesimal numbers to form

. In nonstandard analysis, continuity can be defined as follows.

from the reals to the reals is continuous if its natural extension to the hyperreals has the property that for real x and infinitesimal dx, ƒ(x+

In other words, an infinitesimal increment of the independent variable corresponds to an infinitesimal change of the dependent variable, giving a modern expression to

's definition of continuity.

polynomial functions are continuous. If a function has a domain which is not an interval, the notion of a continuous function as one whose graph you can draw without taking your pencil off the paper is not quite correct. Consider the functions f(x) = 1/x and g(x) = (sin x)/x. Neither function is defined

= 0, so each has domain R \ 0 of real numbers except 0, and each function is continuous. The question of continuity at x = 0 does not arise, since

nor in the domain of g. The function f cannot be extended to a continuous function whose domain is R, since no matter what value is assigned at 0, the resulting function will not be continuous. On the other hand, since the limit of extended continuously to R by defining its value at 0 to be 1.

exponential functions, logarithms, square root function, trigonometric functionsfunction are continuous. Rational functions, however, are not necessarily

R. An example of a rational continuous function is f(x)=1

⁄x-2. The question of continuity at 2 does not arise, since x = 2 is not in the domain of f. An example of a discontinuous function is the function f defined by

0. Pick for instance ε = 1⁄2. There is no δ-neighborhood around ) values to be within ε of f(0). Intuitively we can think of this type of

discontinuity as a sudden jump in function values. Another example of a discontinuous function is the signum or sign function.A more complicated example of a discontinuous function is Thomae's function

is nowhere continuous.

] Facts about continuous functions

infinitesimals mathematically rigorous. The real line is augmented by the addition of infinite and infinitesimal numbers to form

. In nonstandard analysis, continuity can be defined as follows.

from the reals to the reals is continuous if its natural extension to the +dx) − ƒ(x) is

In other words, an infinitesimal increment of the independent variable corresponds to an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis

If a function has a domain which is not an interval, the notion of a continuous function as one whose graph you can draw without taking your pencil off the paper is not quite

. Neither function is defined except 0, and each function is

= 0 does not arise, since x = 0 is neither in the cannot be extended to a continuous

what value is assigned at 0, the resulting function will not be continuous. On the other hand, since the limit of g at 0 is 1, g can be

trigonometric functions and , however, are not necessarily

. The question of continuity at x=

defined by f(x) = 1 if x > 0, f(x) neighborhood around x = 0 that will

(0). Intuitively we can think of this type of

or sign function. Thomae's function.

If two functions f and g are continuous, then possible points x of discontinuity of such x does not belong to the domain of the function domain, or - in other words - is continuous.)

The composition f o g of two continuous functions is continuous.

If a function is differentiable at some point converse is not true: a function that is continuous at for instance the absolute value

[edit] Intermediate value theorem

The intermediate value theoremcompleteness, and states:

If the real-valued function number between f(a) and

For example, if a child grows from 1some time between two and six years of age, the child's height must have been 1.25

As a consequence, if f is continuous on [c in [a, b], f(c) must equal zero

[edit] Extreme value theorem

The extreme value theorem states that if a function closed and bounded set) and is continuous there, then the function attexists c ∈ [a,b] with f(c) ≥ f(x) for all statements are not, in general, true if the function is defined on an open interval (that is not both closed and bounded), as, for example, the continuous function on the open interval (0,1), does not attain a maximum, being unbounded above.

[edit] Directional continuity

A right continuous functionA left continuous function

A function may happen to be continuous in only one direction, either from the "left" or from the "right". A right-continuous function is a function which is continuous at all points when

are continuous, then f + g, fg, and f/g are continuous. (Note. The only of discontinuity of f/g are the solutions of the equation g(x

does not belong to the domain of the function f/g. Hence f/g is continuous on its entire is continuous.)

of two continuous functions is continuous.

at some point c of its domain, then it is also continuous at converse is not true: a function that is continuous at c need not be differentiable there. Consider

absolute value function at c = 0.

] Intermediate value theorem

intermediate value theorem is an existence theorem, based on the real number property of

valued function f is continuous on the closed interval [a, ) and f(b), then there is some number c in [a, b] 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

is continuous on [a, b] and f(a) and f(b) differ in signzero.

] Extreme value theorem

states that if a function f is defined on a closed interval [closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there

) for all x ∈ [a,b]. The same is true of the minimum of statements are not, in general, true if the function is defined on an open interval (that is not both closed and bounded), as, for example, the continuous function on the open interval (0,1), does not attain a maximum, being unbounded above.

] Directional continuity

A left continuous function

A function may happen to be continuous in only one direction, either from the "left" or from the function is a function which is continuous at all points when

are continuous. (Note. The only x) = 0; but then any

is continuous on its entire

of its domain, then it is also continuous at c. The need not be differentiable there. Consider

, based on the real number property of

b] and k is some ] such that f(c) = k.

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.

sign, then, at some point

is defined on a closed interval [a,b] (or any ains its maximum, i.e. there

]. 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.

A function may happen to be continuous in only one direction, either from the "left" or from the function is a function which is continuous at all points when

approached from the right. Technically, the formal definition is similar to the definition above for a continuous function but modified as follows:

The function ƒ 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 ƒ(x) will satisfy

Notice that x must be larger than c, that is on the right of c. If x were also allowed to take values less than c, this would be the definition of continuity. This restriction makes it possible for the function to have a discontinuity at c, but still be right continuous at c, as pictured.

Likewise a left-continuous function is a function which is continuous at all points when approached from the left, that is, c − δ < x < c.

A function is continuous if and only if it is both right-continuous and left-continuous.

[edit] Continuous functions between metric spaces

Now consider a function f from one metric space (X, dX) to another metric space (Y, dY). Then f is continuous at the point c in X 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.

This can also be formulated in terms of sequences and limits: the function f is continuous at the point c if for every sequence (xn) in X with limit lim xn = c, we have lim f(xn) = f(c). Continuous functions transform limits into limits.

This 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. Continuous functions transform convergent sequences into Cauchy sequences.

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.

[edit] Continuous functions between topological spaces

Continuity of a function at a point

The above definitions of continuous functions can be generalized to functions from one topological space to another in a natural way; a function spaces, is continuous if and only if

is open.

However, this definition is often difficult to use directly. Instead, suppose we have a function from X to Y, where X, Y are topological spaces. We say any neighborhood V of f(x), there is a neighborhood definition appears complex, the intuition is that no matter how "small" always find a U containing x that will map inside it. If simply say f is continuous.

In a metric space, it is equivalent to consider the x and f(x) instead of all neighborhoods. This leads to the standard function from real analysis, which says roughly that a function is continuous if all points close to x map to points close to f(x). This only really makes sense in a metric a notion of distance.

Note, however, that if the target space is only if the limit of f as x approaches

[edit] Definitions

Several equivalent definitions for a topological structureequivalent ways to define a continuous function.

[edit] Open and closed set definition

Continuity of a function at a point

The above definitions of continuous functions can be generalized to functions from one to another in a natural way; a function f : X → Y, where X

if and only if for every open set V ⊆ Y, the inverse image

However, this definition is often difficult to use directly. Instead, suppose we have a function are topological spaces. We say f is continuous at x

), there is a neighborhood U of x such that f(U) ⊆definition appears complex, the intuition is that no matter how "small" V becomes, we can

that will map inside it. If f is continuous at every

, it is equivalent to consider the neighbourhood system of ad of all neighborhoods. This leads to the standard ε-δ definition of a continuous

function from real analysis, which says roughly that a function is continuous if all points close to ). This only really makes sense in a metric space, however, which has

Note, however, that if the target space is Hausdorff, it is still true that f is continuous at approaches a is f(a). At an isolated point, every function is continuous.

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

] Open and closed set definition

The above definitions of continuous functions can be generalized to functions from one X and Y are topological

, the inverse image

However, this definition is often difficult to use directly. Instead, suppose we have a function f x for some x ∈ X if for ⊆ V. Although this becomes, we can

is continuous at every x ∈ X, then we

of open balls centered at definition of a continuous

function from real analysis, which says roughly that a function is continuous if all points close to space, however, which has

is continuous at a if and . At an isolated point, every function is continuous.

exist and thus there are several

The most common notion of continuifunctions for which the preimagesset formulation is the closed set formulationclosed sets are closed.

[edit] Neighborhood definition

Definitions based on preimages are often difffunction f : X → Y, where X and x ∈ X if for any neighborhoodAlthough this definition appears complicated, the intuition is that no matter how "small" becomes, we can always find a x ∈ X, then we simply say f is continuous.

In a metric space, it is equivalent to consider the x and f(x) instead of all neighborhoods. This leads to the standard function from real analysis, which says roughly that a function is continuous if all points close to x map to points close to f(x). This only really makes sense in a metric space, however, which has a notion of distance.

Note, however, that if the target space is only if the limit of f as x approaches

[edit] Sequences and nets

In several contexts, the topology of a space is conveniently specified in terms of many instances, this is accomplished by specifying when a point is the for some spaces that are too large in some sense, one specifies also when a point is the limimore general sets of points indexed by a 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

In detail, a function f : X → Y converges to a limit x, the sequence (

The most common notion of continuity in topology defines continuous functions as those preimages (or inverse images) of open sets are open. Similar to the open

closed set formulation, which says that preimages (or inverse images) of

] Neighborhood definition

Definitions based on preimages are often difficult to use directly. Instead, suppose we have a and Y are topological spaces.[5] We say f is continuous at

neighborhood V of f(x), there is a neighborhood U of x such that Although this definition appears complicated, the intuition is that no matter how "small" becomes, we can always find a U containing x that will map inside it. If f is continuous at every

is continuous.

, it is equivalent to consider the neighbourhood system of ) instead of all neighborhoods. This leads to the standard δ-ε definition of a continuous


Note, however, that if the target space is Hausdorff, it is still true that f is continuous at approaches a is f(a). At an isolated point, every function is continuous.

In several contexts, the topology of a space is conveniently specified in terms of is accomplished by specifying when a point is the limit of a sequence

for some spaces that are too large in some sense, one specifies also when a point is the limimore general sets of points indexed by a directed set, known as nets. A function is 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.

is sequentially continuous if whenever a sequence (, the sequence (f(xn)) converges to f(x). Thus sequentially continuous

ty in topology defines continuous functions as those are open. Similar to the open

, which says that preimages (or inverse images) of

icult to use directly. Instead, suppose we have a continuous at x for some such that f(U) ⊆ V.

Although this definition appears complicated, the intuition is that no matter how "small" V is continuous at every

of open balls centered at definition of a continuous


is continuous at a if and . At an isolated point, every function is continuous.

In several contexts, the topology of a space is conveniently specified in terms of limit points. In 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 . A function is 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

is a necessary and sufficient condition.

if whenever a sequence (xn) in X ). Thus sequentially continuous

functions "preserve sequentialis a first-countable space, then the converse also holds: any function preserving sequential liis continuous. In particular, if equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equThis motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nefunctions.

[edit] Closure operator definition

Given two topological spaces (a function

is continuous if for all subsets

One might therefore suspect that given two topological spaces (and int ' are two interior operators


or perhaps if

however, neither of these conditions is either necessary or sufficient for continuity.

Instead, we must resort to inverse images: given two topological spaces (where int and int ' are two interior operators then a func


functions "preserve sequential limits". Every continuous function is sequentially continuous. If , then the converse also holds: any function preserving sequential li

is continuous. In particular, if X is a metric space, sequential continuity and continuity are countable spaces, sequential continuity might be strictly weaker than

continuity. (The spaces for which the two properties are equivalent are called 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

] Closure operator definition

topological spaces (X,cl) and (X ' ,cl ') where cl and cl ' are two closure operators

if for all subsets A of X

that given two topological spaces (X,int) and (interior operators then a function

if for all subsets A of X


Instead, we must resort to inverse images: given two topological spaces (X,int) and (' are two interior operators then a function

if for all subsets A of X '

limits". Every continuous function is sequentially continuous. If X , then the converse also holds: any function preserving sequential limits

is a metric space, sequential continuity and continuity are countable spaces, sequential continuity might be strictly weaker than

ivalent are called sequential spaces.) This motivates the consideration of nets instead of sequences in general topological spaces.

ts, and in fact this property characterizes continuous

closure operators then

,int) and (X ' ,int ') where int


X,int) and (X ' ,int ')

We can also write that given two topological spaces (closure operators then a function


[edit] Closeness relation definition

Given two topological spaces (function

is continuous if for all points

This is another way of writing the closure operator definition.

[edit] Useful properties of continuous maps

Some facts about continuous maps between topological

• If f : X → Y and g : Y →• If f : X → Y is continuous and

o X is compact, then o X is connected, then o X is path-connectedo X is Lindelöf, then o X is separable, then

• The identity map idX : (comparison of topologies

[edit] Other notes

If a set is given the discrete topologythe domain set is given the indiscrete topologycontinuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous.

Given a set X, a partial orderingfunction between two topological

We can also write that given two topological spaces (X,cl) and (X ' ,cl ') where cl and clclosure operators then a function

if for all subsets A of X '

] Closeness relation definition

Given two topological spaces (X,δ) and (X' ,δ') where δ and δ' are two closeness relations

if for all points x and of X and all subsets A of X,

This is another way of writing the closure operator definition.

] Useful properties of continuous maps

Some facts about continuous maps between topological spaces:

→ Z are continuous, then so is the composition is continuous and

, then f(X) is compact. , then f(X) is connected.

connected, then f(X) is path-connected. , then f(X) is Lindelöf. , then f(X) is separable. : (X, τ2) → (X, τ1) is continuous if and only if τ

topologies).

discrete topology, all functions with that space as a domain are continuous. If indiscrete topology and the range set is at least

continuous functions are the constant functions. Conversely, any function whose range is

partial ordering can be defined on the possible topologies on function between two topological spaces stays continuous if we strengthen

') where cl and cl ' are two

closeness relations then a

are continuous, then so is the composition g ∘ f : X → Z.

) is continuous if and only if τ1 ⊆ τ2 (see also

, all functions with that space as a domain are continuous. If T0, then the only

continuous functions are the constant functions. Conversely, any function whose range is

on X. A continuous the topology of the

domain space or weaken the topology of the codomain space. Thus we can consider the continuity of a given function a topological property, depending only on the topologies of its domain and codomain spaces.

For a function f from a topological space X to a set S, one defines the final topology on S 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 which makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f. This construction can be generalized to an arbitrary family of functions X → S.

Dually, for a function f from a set S to a topological space, one defines the initial topology on S by letting the open sets of S be those subsets A of S for which f(A) is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus the initial topology can be characterized as the coarsest topology on S which makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X. This construction can be generalized to an arbitrary family of functions S → X.

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, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open.

If a function is a bijection, then it has an inverse function. The inverse of a continuous bijection is open, but need not be continuous. If it is, this special function is called a homeomorphism. If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is automatically a homeomorphism.

[edit] Continuous functions between partially ordered sets

In order theory, continuity of a function between posets is Scott continuity. Let X be a complete lattice, then a function f : X → X is continuous if, for each subset Y of X, we have sup f(Y) = f(sup Y).

[edit] Continuous binary relation

A binary relation R on A is continuous if R(a, b) whenever there are sequences (ak)i and (bk)i in A which converge to a and b respectively for which R(ak, bk) for all k. Clearly, if one treats R as a characteristic function in two variables, this definition of continuous is identical to that for continuous functions.

[edit] Continuity space

A continuity space[6][7] 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

[edit] See also

• Absolute continuity • Bounded linear operator• Classification of discontinuities• Coarse function • Continuous functor • Continuous stochastic process• Dini continuity • Discrete function • Equicontinuity • Lipschitz continuity • Normal function • Piecewise • Scott continuity • Semicontinuity • Smooth function • Symmetrically continuous function• Uniform continuity

[edit] Notes

Wikimedia Commons has media related to:

1. ^ Grabiner, Judith V. (March 1983). Origins of Rigorous Calculus"doi:10.2307/2975545.

2. ^ Introduction to Real Analysisp. 172

3. ^ Introduction to Real AnalysisAdvanced Look at the Existence of the Proper Riemann Integral", pp. 171

4. ^ http://www.math.wisc.edu/~keisler/calc.html5. ^ f is a function f : X →

function f is defined on the elements of the set However continuity of the function does depend on the topologies used.

6. ^ Quantales and continuity spaces7. ^ All topologies come from genera

Monthly, 1988 8. ^ Continuity spaces: Reconciling domains and metric spaces, B Flagg, R Kopperman

Theoretical Computer Science,

is a generalization of metric spaces and posets, which uses the concept of , and that can be used to unify the notions of metric spaces and domains

Bounded linear operator Classification of discontinuities

Continuous stochastic process

Symmetrically continuous function

Wikimedia Commons has media related to: Continuity (functions)

V. (March 1983). "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus". The American Mathematical Monthly

. http://www.maa.org/pubs/Calc_articles/ma002.pdfIntroduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2,

Introduction to Real Analysis, updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171

http://www.math.wisc.edu/~keisler/calc.html → Y between two topological spaces (X,TX) and (

is defined on the elements of the set X, not on the elements of the topology However continuity of the function does depend on the topologies used.

Quantales and continuity spaces, RC Flagg - Algebra Universalis, 1997All topologies come from generalized metrics, R Kopperman - American Mathematical

Continuity spaces: Reconciling domains and metric spaces, B Flagg, R Kopperman Theoretical Computer Science, 1997

is a generalization of metric spaces and posets, which uses the concept of domains.[8]

"Who Gave You the Epsilon? Cauchy and the American Mathematical Monthly 90 (3): 185–194.

http://www.maa.org/pubs/Calc_articles/ma002.pdf. updated April 2010, William F. Trench, Theorem 3.5.2,

updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177

) and (Y,TY). That is, the , not on the elements of the topology TX.

However continuity of the function does depend on the topologies used. Algebra Universalis, 1997

American Mathematical

Continuity spaces: Reconciling domains and metric spaces, B Flagg, R Kopperman -