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Wikipedia

Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is, a change without jump) of the argument induces a continuous variation of the value of the function. This means 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account at time t would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.

History edit

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 edit

Definition edit

 
The function   is continuous on its domain ( ), but discontinuous (not-continuous or singularity) at  [7].Nevertheless, the Cauchy principal value can be defined. On the other hand, in complex analysis ( , especially  .), this point (x=0) is not regarded as "undefined" and it is called a singularity, because when thinking of   as a complex variable, this point is a pole of order one, and then Laurent series with at most finite principal part can be defined around the singular points. Also, the Riemann sphere is often used as a model to study functions like the example.

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 are the functions   and   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 edit

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 edit

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 edit

 
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 edit

 
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 edit

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 edit

 
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 edit

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 edit

 
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 edit

 
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.

Properties edit

A useful lemma edit

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 edit

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 edit

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 edit

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 edit

 
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 edit

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 edit

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 edit

 
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 edit

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 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 edit

 
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 edit

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

Sequences and nets edit

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

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 edit

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
continuous, function, mathematics, continuous, function, function, such, that, continuous, variation, that, change, without, jump, argument, induces, continuous, variation, value, function, this, means, there, abrupt, changes, value, known, discontinuities, mo. In mathematics a continuous function is a function such that a continuous variation that is a change without jump of the argument induces a continuous variation of the value of the function This means 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 Contents 1 History 2 Real functions 2 1 Definition 2 1 1 Definition in terms of limits of functions 2 1 2 Definition in terms of neighborhoods 2 1 3 Definition in terms of limits of sequences 2 1 4 Weierstrass and Jordan definitions epsilon delta of continuous functions 2 1 5 Definition in terms of control of the remainder 2 1 6 Definition using oscillation 2 1 7 Definition using the hyperreals 2 2 Construction of continuous functions 2 3 Examples of discontinuous functions 2 4 Properties 2 4 1 A useful lemma 2 4 2 Intermediate value theorem 2 4 3 Extreme value theorem 2 4 4 Relation to differentiability and integrability 2 4 5 Pointwise and uniform limits 2 5 Directional and semi continuity 3 Continuous functions between metric spaces 3 1 Uniform Holder and Lipschitz continuity 4 Continuous functions between topological spaces 4 1 Continuity at a point 4 2 Alternative definitions 4 2 1 Sequences and nets 4 2 2 Closure operator and interior operator definitions 4 2 3 Filters and prefilters 4 3 Properties 4 4 Homeomorphisms 4 5 Defining topologies via continuous functions 5 Related notions 6 See also 7 References 8 BibliographyHistory editA form of the epsilon delta definition of continuity was first given by Bernard Bolzano in 1817 Augustin Louis Cauchy defined continuity of y f x displaystyle y f x nbsp as follows an infinitely small increment a displaystyle alpha nbsp of the independent variable x always produces an infinitely small change f x a f x displaystyle f x alpha f x nbsp 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 Edouard 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 editDefinition edit nbsp The function f x 1 x displaystyle f x tfrac 1 x nbsp is continuous on its domain R 0 displaystyle mathbb R setminus 0 nbsp but discontinuous not continuous or singularity at x 0 displaystyle x 0 nbsp 7 Nevertheless the Cauchy principal value can be defined On the other hand in complex analysis C displaystyle mathbb C nbsp especially C displaystyle widehat mathbb C nbsp this point x 0 is not regarded as undefined and it is called a singularity because when thinking of x displaystyle x nbsp as a complex variable this point is a pole of order one and then Laurent series with at most finite principal part can be defined around the singular points Also the Riemann sphere is often used as a model to study functions like the example 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 f x displaystyle f x nbsp as x tends to c is equal to f c displaystyle f c nbsp 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 displaystyle infty infty nbsp 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 f x x displaystyle f x sqrt x nbsp is continuous on its whole domain which is the closed interval 0 displaystyle 0 infty nbsp Many commonly encountered functions are partial functions that have a domain formed by all real numbers except some isolated points Examples are the functions x 1 x textstyle x mapsto frac 1 x nbsp and x tan x displaystyle x mapsto tan x nbsp 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 x 1 x textstyle x mapsto frac 1 x nbsp and x sin 1 x textstyle x mapsto sin frac 1 x nbsp 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 Letf D R displaystyle f D to mathbb R nbsp be a function defined on a subset D displaystyle D nbsp of the set R displaystyle mathbb R nbsp of real numbers This subset D displaystyle D nbsp is the domain of f Some possible choices include D R displaystyle D mathbb R nbsp i e D displaystyle D nbsp is the whole set of real numbers or for a and b real numbers D a b x R a x b displaystyle D a b x in mathbb R mid a leq x leq b nbsp D displaystyle D nbsp is a closed interval or D a b x R a lt x lt b displaystyle D a b x in mathbb R mid a lt x lt b nbsp D displaystyle D nbsp is an open interval In the case of the domain D displaystyle D nbsp being defined as an open interval a displaystyle a nbsp and b displaystyle b nbsp do not belong to D displaystyle D nbsp and the values of f a displaystyle f a nbsp and f b displaystyle f b nbsp do not matter for continuity on D displaystyle D nbsp Definition in terms of limits of functions edit The function f is continuous at some point c of its domain if the limit of f x displaystyle f x nbsp as x approaches c through the domain of f exists and is equal to f c displaystyle f c nbsp 9 In mathematical notation this is written aslim x c f x f c displaystyle lim x to c f x f c nbsp 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 f c displaystyle f c nbsp Here we have assumed that the domain of f does not have any isolated points Definition in terms of neighborhoods edit 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 displaystyle f c nbsp as the width of the neighborhood around c shrinks to zero More precisely a function f is continuous at a point c of its domain if for any neighborhood N 1 f c displaystyle N 1 f c nbsp there is a neighborhood N 2 c displaystyle N 2 c nbsp in its domain such that f x N 1 f c displaystyle f x in N 1 f c nbsp whenever x N 2 c displaystyle x in N 2 c nbsp 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 edit nbsp The sequence exp 1 n converges to exp 0 1One can instead require that for any sequence x n n N displaystyle x n n in mathbb N nbsp of points in the domain which converges to c the corresponding sequence f x n n N displaystyle left f x n right n in mathbb N nbsp converges to f c displaystyle f c nbsp In mathematical notation x n n N D lim n x n c lim n f x n f c displaystyle forall x n n in mathbb N subset D lim n to infty x n c Rightarrow lim n to infty f x n f c nbsp Weierstrass and Jordan definitions epsilon delta of continuous functions edit nbsp Illustration of the e d definition at x 2 any value d 0 5 satisfies the condition of the definition for e 0 5 Explicitly including the definition of the limit of a function we obtain a self contained definition Given a function f D R displaystyle f D to mathbb R nbsp as above and an element x 0 displaystyle x 0 nbsp of the domain D displaystyle D nbsp f displaystyle f nbsp is said to be continuous at the point x 0 displaystyle x 0 nbsp when the following holds For any positive real number e gt 0 displaystyle varepsilon gt 0 nbsp however small there exists some positive real number d gt 0 displaystyle delta gt 0 nbsp such that for all x displaystyle x nbsp in the domain of f displaystyle f nbsp with x 0 d lt x lt x 0 d displaystyle x 0 delta lt x lt x 0 delta nbsp the value of f x displaystyle f x nbsp satisfiesf x 0 e lt f x lt f x 0 e displaystyle f left x 0 right varepsilon lt f x lt f x 0 varepsilon nbsp Alternatively written continuity of f D R displaystyle f D to mathbb R nbsp at x 0 D displaystyle x 0 in D nbsp means that for every e gt 0 displaystyle varepsilon gt 0 nbsp there exists a d gt 0 displaystyle delta gt 0 nbsp such that for all x D displaystyle x in D nbsp x x 0 lt d implies f x f x 0 lt e displaystyle left x x 0 right lt delta text implies f x f x 0 lt varepsilon nbsp More intuitively we can say that if we want to get all the f x displaystyle f x nbsp values to stay in some small neighborhood around f x 0 displaystyle f left x 0 right nbsp we need to choose a small enough neighborhood for the x displaystyle x nbsp values around x 0 displaystyle x 0 nbsp If we can do that no matter how small the f x 0 displaystyle f x 0 nbsp neighborhood is then f displaystyle f nbsp is continuous at x 0 displaystyle x 0 nbsp In modern terms this is generalized by the definition of continuity of a function with respect to a basis for the topology here the metric topology Weierstrass had required that the interval x 0 d lt x lt x 0 d displaystyle x 0 delta lt x lt x 0 delta nbsp be entirely within the domain D displaystyle D nbsp but Jordan removed that restriction Definition in terms of control of the remainder edit 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 C 0 0 displaystyle C 0 infty to 0 infty nbsp is called a control function if C is non decreasing inf d gt 0 C d 0 displaystyle inf delta gt 0 C delta 0 nbsp A function f D R displaystyle f D to R nbsp is C continuous at x 0 displaystyle x 0 nbsp if there exists such a neighbourhood N x 0 textstyle N x 0 nbsp that f x f x 0 C x x 0 for all x D N x 0 displaystyle f x f x 0 leq C left left x x 0 right right text for all x in D cap N x 0 nbsp A function is continuous in x 0 displaystyle x 0 nbsp 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 C displaystyle mathcal C nbsp a function is C displaystyle mathcal C nbsp continuous if it is C displaystyle C nbsp continuous for some C C displaystyle C in mathcal C nbsp For example the Lipschitz and Holder continuous functions of exponent a below are defined by the set of control functionsC L i p s c h i t z C C d K d K gt 0 displaystyle mathcal C mathrm Lipschitz C C delta K delta K gt 0 nbsp respectively C Holder a C C d K d a K gt 0 displaystyle mathcal C text Holder alpha C C delta K delta alpha K gt 0 nbsp Definition using oscillation edit nbsp 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 x 0 displaystyle x 0 nbsp if and only if its oscillation at that point is zero 10 in symbols w f x 0 0 displaystyle omega f x 0 0 nbsp 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 e displaystyle varepsilon nbsp hence a G d displaystyle G delta nbsp set and gives a rapid proof of one direction of the Lebesgue integrability condition 11 The oscillation is equivalent to the e d displaystyle varepsilon delta nbsp 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 e 0 displaystyle varepsilon 0 nbsp there is no d displaystyle delta nbsp that satisfies the e d displaystyle varepsilon delta nbsp definition then the oscillation is at least e 0 displaystyle varepsilon 0 nbsp and conversely if for every e displaystyle varepsilon nbsp there is a desired d displaystyle delta nbsp 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 edit 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 f x d x f x displaystyle f x dx f x nbsp 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 edit nbsp 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 Givenf g D R displaystyle f g colon D to mathbb R nbsp then the sum of continuous functions s f g displaystyle s f g nbsp defined by s x f x g x displaystyle s x f x g x nbsp for all x D displaystyle x in D nbsp is continuous in D displaystyle D nbsp The same holds for the product of continuous functions p f g displaystyle p f cdot g nbsp defined by p x f x g x displaystyle p x f x cdot g x nbsp for all x D displaystyle x in D nbsp is continuous in D displaystyle D nbsp Combining the above preservations of continuity and the continuity of constant functions and of the identity function I x x displaystyle I x x nbsp on R displaystyle mathbb R nbsp one arrives at the continuity of all polynomial functions on R displaystyle mathbb R nbsp such asf x x 3 x 2 5 x 3 displaystyle f x x 3 x 2 5x 3 nbsp pictured on the right nbsp The graph of a continuous rational function The function is not defined for x 2 displaystyle x 2 nbsp The vertical and horizontal lines are asymptotes In the same way it can be shown that the reciprocal of a continuous functionr 1 f displaystyle r 1 f nbsp defined by r x 1 f x displaystyle r x 1 f x nbsp for all x D displaystyle x in D nbsp such that f x 0 displaystyle f x neq 0 nbsp is continuous in D x f x 0 displaystyle D setminus x f x 0 nbsp This implies that excluding the roots of g displaystyle g nbsp the quotient of continuous functionsq f g displaystyle q f g nbsp defined by q x f x g x displaystyle q x f x g x nbsp for all x D displaystyle x in D nbsp such that g x 0 displaystyle g x neq 0 nbsp is also continuous on D x g x 0 displaystyle D setminus x g x 0 nbsp For example the function pictured y x 2 x 1 x 2 displaystyle y x frac 2x 1 x 2 nbsp is defined for all real numbers x 2 displaystyle x neq 2 nbsp and is continuous at every such point Thus it is a continuous function The question of continuity at x 2 displaystyle x 2 nbsp does not arise since x 2 displaystyle x 2 nbsp is not in the domain of y displaystyle y nbsp There is no continuous function F R R displaystyle F mathbb R to mathbb R nbsp that agrees with y x displaystyle y x nbsp for all x 2 displaystyle x neq 2 nbsp nbsp The sinc and the cos functionsSince the function sine is continuous on all reals the sinc function G x sin x x displaystyle G x sin x x nbsp is defined and continuous for all real x 0 displaystyle x neq 0 nbsp However unlike the previous example G can be extended to a continuous function on all real numbers by defining the value G 0 displaystyle G 0 nbsp to be 1 which is the limit of G x displaystyle G x nbsp when x approaches 0 i e G 0 lim x 0 sin x x 1 displaystyle G 0 lim x to 0 frac sin x x 1 nbsp Thus by setting G x sin x x if x 0 1 if x 0 displaystyle G x begin cases frac sin x x amp text if x neq 0 1 amp text if x 0 end cases nbsp 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 functionsg D g R R g R and f D f R R f D g displaystyle g D g subseteq mathbb R to R g subseteq mathbb R quad text and quad f D f subseteq mathbb R to R f subseteq D g nbsp their composition denoted as c g f D f R displaystyle c g circ f D f to mathbb R nbsp and defined by c x g f x displaystyle c x g f x nbsp is continuous This construction allows stating for example thate sin ln x displaystyle e sin ln x nbsp is continuous for all x gt 0 displaystyle x gt 0 nbsp Examples of discontinuous functions edit nbsp Plot of the signum function It shows that lim n sgn 1 n sgn lim n 1 n displaystyle lim n to infty operatorname sgn left tfrac 1 n right neq operatorname sgn left lim n to infty tfrac 1 n right nbsp Thus the signum function is discontinuous at 0 see section 2 1 3 An example of a discontinuous function is the Heaviside step function H displaystyle H nbsp defined byH x 1 if x 0 0 if x lt 0 displaystyle H x begin cases 1 amp text if x geq 0 0 amp text if x lt 0 end cases nbsp Pick for instance e 1 2 displaystyle varepsilon 1 2 nbsp Then there is no d displaystyle delta nbsp neighborhood around x 0 displaystyle x 0 nbsp i e no open interval d d displaystyle delta delta nbsp with d gt 0 displaystyle delta gt 0 nbsp that will force all the H x displaystyle H x nbsp values to be within the e displaystyle varepsilon nbsp neighborhood of H 0 displaystyle H 0 nbsp i e within 1 2 3 2 displaystyle 1 2 3 2 nbsp Intuitively we can think of this type of discontinuity as a sudden jump in function values Similarly the signum or sign functionsgn x 1 if x gt 0 0 if x 0 1 if x lt 0 displaystyle operatorname sgn x begin cases 1 amp text if x gt 0 0 amp text if x 0 1 amp text if x lt 0 end cases nbsp is discontinuous at x 0 displaystyle x 0 nbsp but continuous everywhere else Yet another example the function f x sin x 2 if x 0 0 if x 0 displaystyle f x begin cases sin left x 2 right amp text if x neq 0 0 amp text if x 0 end cases nbsp is continuous everywhere apart from x 0 displaystyle x 0 nbsp nbsp 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 f x 1 if x 0 1 q if x p q in lowest terms is a rational number 0 if x is irrational displaystyle f x begin cases 1 amp text if x 0 frac 1 q amp text if x frac p q text in lowest terms is a rational number 0 amp text if x text is irrational end cases nbsp is continuous at all irrational numbers and discontinuous at all rational numbers In a similar vein Dirichlet s function the indicator function for the set of rational numbers D x 0 if x is irrational R Q 1 if x is rational Q displaystyle D x begin cases 0 amp text if x text is irrational in mathbb R setminus mathbb Q 1 amp text if x text is rational in mathbb Q end cases nbsp is nowhere continuous Properties edit A useful lemma edit Let f x displaystyle f x nbsp be a function that is continuous at a point x 0 displaystyle x 0 nbsp and y 0 displaystyle y 0 nbsp be a value such f x 0 y 0 displaystyle f left x 0 right neq y 0 nbsp Then f x y 0 displaystyle f x neq y 0 nbsp throughout some neighbourhood of x 0 displaystyle x 0 nbsp 13 Proof By the definition of continuity take e y 0 f x 0 2 gt 0 displaystyle varepsilon frac y 0 f x 0 2 gt 0 nbsp then there exists d gt 0 displaystyle delta gt 0 nbsp such that f x f x 0 lt y 0 f x 0 2 whenever x x 0 lt d displaystyle left f x f x 0 right lt frac left y 0 f x 0 right 2 quad text whenever quad x x 0 lt delta nbsp Suppose there is a point in the neighbourhood x x 0 lt d displaystyle x x 0 lt delta nbsp for which f x y 0 displaystyle f x y 0 nbsp then we have the contradiction f x 0 y 0 lt f x 0 y 0 2 displaystyle left f x 0 y 0 right lt frac left f x 0 y 0 right 2 nbsp Intermediate value theorem edit 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 displaystyle a b nbsp and k is some number between f a displaystyle f a nbsp and f b displaystyle f b nbsp then there is some number c a b displaystyle c in a b nbsp such that f c k displaystyle f c k nbsp 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 displaystyle a b nbsp and f a displaystyle f a nbsp and f b displaystyle f b nbsp differ in sign then at some point c a b displaystyle c in a b nbsp f c displaystyle f c nbsp must equal zero Extreme value theorem edit The extreme value theorem states that if a function f is defined on a closed interval a b displaystyle a b nbsp or any closed and bounded set and is continuous there then the function attains its maximum i e there exists c a b displaystyle c in a b nbsp with f c f x displaystyle f c geq f x nbsp for all x a b displaystyle x in a b nbsp 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 displaystyle a b nbsp or any set that is not both closed and bounded as for example the continuous function f x 1 x displaystyle f x frac 1 x nbsp defined on the open interval 0 1 does not attain a maximum being unbounded above Relation to differentiability and integrability edit Every differentiable functionf a b R displaystyle f a b to mathbb R nbsp is continuous as can be shown The converse does not hold for example the absolute value function f x x x if x 0 x if x lt 0 displaystyle f x x begin cases x amp text if x geq 0 x amp text if x lt 0 end cases nbsp is everywhere continuous However it is not differentiable at x 0 displaystyle x 0 nbsp but is so everywhere else Weierstrass s function is also everywhere continuous but nowhere differentiable The derivative f x of a differentiable function f x need not be continuous If f x is continuous f x is said to be continuously differentiable The set of such functions is denoted C 1 a b displaystyle C 1 a b nbsp More generally the set of functionsf W R displaystyle f Omega to mathbb R nbsp from an open interval or open subset of R displaystyle mathbb R nbsp W displaystyle Omega nbsp to the reals such that f is n displaystyle n nbsp times differentiable and such that the n displaystyle n nbsp th derivative of f is continuous is denoted C n W displaystyle C n Omega nbsp See differentiability class In the field of computer graphics properties related but not identical to C 0 C 1 C 2 displaystyle C 0 C 1 C 2 nbsp are sometimes called G 0 displaystyle G 0 nbsp continuity of position G 1 displaystyle G 1 nbsp continuity of tangency and G 2 displaystyle G 2 nbsp continuity of curvature see Smoothness of curves and surfaces Every continuous functionf a b R displaystyle f a b to mathbb R nbsp 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 edit nbsp A sequence of continuous functions f n x displaystyle f n x nbsp whose pointwise limit function f x displaystyle f x nbsp is discontinuous The convergence is not uniform Given a sequencef 1 f 2 I R displaystyle f 1 f 2 dotsc I to mathbb R nbsp of functions such that the limit f x lim n f n x displaystyle f x lim n to infty f n x nbsp exists for all x D displaystyle x in D nbsp the resulting function f x displaystyle f x nbsp is referred to as the pointwise limit of the sequence of functions f n n N displaystyle left f n right n in N nbsp The pointwise limit function need not be continuous even if all functions f n displaystyle f n nbsp are continuous as the animation at the right shows However f is continuous if all functions f n displaystyle f n nbsp 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 edit nbsp A right continuous function nbsp A left continuous function Discontinuous functions may be discontinuous in a restricted way giving rise to the concept of directional continuity or right and left continuous functions and semi continuity Roughly speaking a function is right continuous if no jump occurs when the limit point is approached from the right Formally f is said to be right continuous at the point c if the following holds For any number e gt 0 displaystyle varepsilon gt 0 nbsp however small there exists some number d gt 0 displaystyle delta gt 0 nbsp such that for all x in the domain with c lt x lt c d displaystyle c lt x lt c delta nbsp the value of f x displaystyle f x nbsp will satisfy f x f c lt e displaystyle f x f c lt varepsilon nbsp 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 c d lt x lt c displaystyle c delta lt x lt c nbsp 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 e gt 0 displaystyle varepsilon gt 0 nbsp there exists some number d gt 0 displaystyle delta gt 0 nbsp such that for all x in the domain with x c lt d displaystyle x c lt delta nbsp the value of f x displaystyle f x nbsp satisfiesf x f c ϵ displaystyle f x geq f c epsilon nbsp The reverse condition is upper semi continuity Continuous functions between metric spaces editThe concept of continuous real valued functions can be generalized to functions between metric spaces A metric space is a set X displaystyle X nbsp equipped with a function called metric d X displaystyle d X nbsp that can be thought of as a measurement of the distance of any two elements in X Formally the metric is a functiond X X X R displaystyle d X X times X to mathbb R nbsp that satisfies a number of requirements notably the triangle inequality Given two metric spaces X d X displaystyle left X d X right nbsp and Y d Y displaystyle left Y d Y right nbsp and a function f X Y displaystyle f X to Y nbsp then f displaystyle f nbsp is continuous at the point c X displaystyle c in X nbsp with respect to the given metrics if for any positive real number e gt 0 displaystyle varepsilon gt 0 nbsp there exists a positive real number d gt 0 displaystyle delta gt 0 nbsp such that all x X displaystyle x in X nbsp satisfying d X x c lt d displaystyle d X x c lt delta nbsp will also satisfy d Y f x f c lt e displaystyle d Y f x f c lt varepsilon nbsp As in the case of real functions above this is equivalent to the condition that for every sequence x n displaystyle left x n right nbsp in X displaystyle X nbsp with limit lim x n c displaystyle lim x n c nbsp we have lim f x n f c displaystyle lim f left x n right f c nbsp The latter condition can be weakened as follows f displaystyle f nbsp is continuous at the point c displaystyle c nbsp if and only if for every convergent sequence x n displaystyle left x n right nbsp in X displaystyle X nbsp with limit c displaystyle c nbsp the sequence f x n displaystyle left f left x n right right nbsp is a Cauchy sequence and c displaystyle c nbsp is in the domain of f displaystyle f nbsp The set of points at which a function between metric spaces is continuous is a G d displaystyle G delta nbsp set this follows from the e d displaystyle varepsilon delta nbsp definition of continuity This notion of continuity is applied for example in functional analysis A key statement in this area says that a linear operatorT V W displaystyle T V to W nbsp between normed vector spaces V displaystyle V nbsp and W displaystyle W nbsp which are vector spaces equipped with a compatible norm denoted x displaystyle x nbsp is continuous if and only if it is bounded that is there is a constant K displaystyle K nbsp such that T x K x displaystyle T x leq K x nbsp for all x V displaystyle x in V nbsp Uniform Holder and Lipschitz continuity edit nbsp 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 d displaystyle delta nbsp depends on e displaystyle varepsilon nbsp and c in the definition above Intuitively a function f as above is uniformly continuous if the d displaystyle delta nbsp does not depend on the point c More precisely it is required that for every real number e gt 0 displaystyle varepsilon gt 0 nbsp there exists d gt 0 displaystyle delta gt 0 nbsp such that for every c b X displaystyle c b in X nbsp with d X b c lt d displaystyle d X b c lt delta nbsp we have that d Y f b f c lt e displaystyle d Y f b f c lt varepsilon nbsp 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 Holder continuous with exponent a a real number if there is a constant K such that for all b c X displaystyle b c in X nbsp the inequalityd Y f b f c K d X b c a displaystyle d Y f b f c leq K cdot d X b c alpha nbsp holds Any Holder continuous function is uniformly continuous The particular case a 1 displaystyle alpha 1 nbsp is referred to as Lipschitz continuity That is a function is Lipschitz continuous if there is a constant K such that the inequality d Y f b f c K d X b c displaystyle d Y f b f c leq K cdot d X b c nbsp holds for any b c X displaystyle b c in X nbsp 15 The Lipschitz condition occurs for example in the Picard Lindelof theorem concerning the solutions of ordinary differential equations Continuous functions between topological spaces editAnother 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 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 functionf X Y displaystyle f X to Y nbsp between two topological spaces X and Y is continuous if for every open set V Y displaystyle V subseteq Y nbsp the inverse image f 1 V x X f x V displaystyle f 1 V x in X f x in V nbsp is an open subset of X That is f is a function between the sets X and Y not on the elements of the topology T X displaystyle T X nbsp 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 functionsf X T displaystyle f X to T nbsp 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 edit nbsp Continuity at a point For every neighborhood V of f x displaystyle f x nbsp there is a neighborhood U of x such that f U V displaystyle f U subseteq V nbsp The translation in the language of neighborhoods of the e d displaystyle varepsilon delta nbsp definition of continuity leads to the following definition of the continuity at a point A function f X Y displaystyle f X to Y nbsp is continuous at a point x X displaystyle x in X nbsp if and only if for any neighborhood V of f x displaystyle f x nbsp in Y there is a neighborhood U of x displaystyle x nbsp such that f U V displaystyle f U subseteq V nbsp This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using preimages rather than images Also as every set that contains a neighborhood is also a neighborhood and f 1 V displaystyle f 1 V nbsp is the largest subset U of X such that f U V displaystyle f U subseteq V nbsp this definition may be simplified into A function f X Y displaystyle f X to Y nbsp is continuous at a point x X displaystyle x in X nbsp if and only if f 1 V displaystyle f 1 V nbsp is a neighborhood of x displaystyle x nbsp for every neighborhood V of f x displaystyle f x nbsp in Y As an open set is a set that is a neighborhood of all its points a function f X Y displaystyle f X to Y nbsp 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 e d displaystyle varepsilon delta nbsp definition of continuity in the context of metric spaces In general topological spaces there is no notion of nearness or distance If however the target space is a Hausdorff space it is still true that f is continuous at a if and only if the limit of f as x approaches a is f a At an isolated point every function is continuous Given x X displaystyle x in X nbsp a map f X Y displaystyle f X to Y nbsp is continuous at x displaystyle x nbsp if and only if whenever B displaystyle mathcal B nbsp is a filter on X displaystyle X nbsp that converges to x displaystyle x nbsp in X displaystyle X nbsp which is expressed by writing B x displaystyle mathcal B to x nbsp then necessarily f B f x displaystyle f mathcal B to f x nbsp in Y displaystyle Y nbsp If N x displaystyle mathcal N x nbsp denotes the neighborhood filter at x displaystyle x nbsp then f X Y displaystyle f X to Y nbsp is continuous at x displaystyle x nbsp if and only if f N x f x displaystyle f mathcal N x to f x nbsp in Y displaystyle Y nbsp 16 Moreover this happens if and only if the prefilter f N x displaystyle f mathcal N x nbsp is a filter base for the neighborhood filter of f x displaystyle f x nbsp in Y displaystyle Y nbsp 16 Alternative definitions edit Several equivalent definitions for a topological structure exist thus several equivalent ways exist to define a continuous function Sequences and nets edit 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 f X Y displaystyle f X to Y nbsp is sequentially continuous if whenever a sequence x n displaystyle left x n right nbsp in X displaystyle X nbsp converges to a limit x displaystyle x nbsp the sequence f x n displaystyle left f left x n right right nbsp converges to f x displaystyle f x nbsp Thus sequentially continuous functions preserve sequential limits Every continuous function is sequentially continuous If X displaystyle X nbsp 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 displaystyle X nbsp 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 f A R R displaystyle f A subseteq mathbb R to mathbb R nbsp is continuous at x 0 displaystyle x 0 nbsp if and only if it is sequentially continuous at that point ProofProof Assume that f A R R displaystyle f A subseteq mathbb R to mathbb R nbsp is continuous at x 0 displaystyle x 0 nbsp in the sense of ϵ d displaystyle epsilon delta nbsp continuity Let x n n 1 displaystyle left x n right n geq 1 nbsp be a sequence converging at x 0 displaystyle x 0 nbsp such a sequence always exists for example x n x for all n displaystyle x n x text for all n nbsp since f displaystyle f nbsp is continuous at x 0 displaystyle x 0 nbsp ϵ gt 0 d ϵ gt 0 0 lt x x 0 lt d ϵ f x f x 0 lt ϵ displaystyle forall epsilon gt 0 exists delta epsilon gt 0 0 lt x x 0 lt delta epsilon implies f x f x 0 lt epsilon quad nbsp For any such d ϵ displaystyle delta epsilon nbsp we can find a natural number n ϵ gt 0 displaystyle nu epsilon gt 0 nbsp such that for all n gt n ϵ displaystyle n gt nu epsilon nbsp x n x 0 lt d ϵ displaystyle x n x 0 lt delta epsilon nbsp since x n displaystyle left x n right nbsp converges at x 0 displaystyle x 0 nbsp combining this with displaystyle nbsp we obtain ϵ gt 0 n ϵ gt 0 n gt n ϵ f x n f x 0 lt ϵ displaystyle forall epsilon gt 0 exists nu epsilon gt 0 forall n gt nu epsilon quad f x n f x 0 lt epsilon nbsp Assume on the contrary that f displaystyle f nbsp is sequentially continuous and proceed by contradiction suppose f displaystyle f nbsp is not continuous at x 0 displaystyle x 0 nbsp ϵ gt 0 d ϵ gt 0 x d ϵ 0 lt x d ϵ x 0 lt d ϵ f x d ϵ f x 0 gt ϵ displaystyle exists epsilon gt 0 forall delta epsilon gt 0 exists x delta epsilon 0 lt x delta epsilon x 0 lt delta epsilon implies f x delta epsilon f x 0 gt epsilon nbsp then we can take d ϵ 1 n n gt 0 displaystyle delta epsilon 1 n forall n gt 0 nbsp and call the corresponding point x d ϵ x n displaystyle x delta epsilon x n nbsp in this way we have defined a sequence x n n 1 displaystyle x n n geq 1 nbsp such that n gt 0 x n x 0 lt 1 n f x n f x 0 gt ϵ displaystyle forall n gt 0 quad x n x 0 lt frac 1 n quad f x n f x 0 gt epsilon nbsp by construction x n x 0 displaystyle x n to x 0 nbsp but f x n f x 0 displaystyle f x n not to f x 0 nbsp which contradicts the hypothesis of sequentially continuity displaystyle blacksquare nbsp Closure operator and interior operator definitions edit In terms of the interior operator a function f X Y displaystyle f X to Y nbsp between topological spaces is continuous if and only if for every subset B Y displaystyle B subseteq Y nbsp f 1 int Y B int X f 1 B displaystyle f 1 left operatorname int Y B right subseteq operatorname int X left f 1 B right nbsp In terms of the closure operator f X Y displaystyle f X to Y nbsp is continuous if and only if for every subset A X displaystyle A subseteq X nbsp f cl X A cl Y f A displaystyle f left operatorname cl X A right subseteq operatorname cl Y f A nbsp That is to say given any element x X displaystyle x in X nbsp that belongs to the closure of a subset A X displaystyle A subseteq X nbsp f x displaystyle f x nbsp necessarily belongs to the closure of f A displaystyle f A nbsp in span, wikipedia, wiki, book, books, library,

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