Continuous function

 ( Calculus ) 

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on Intuition 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 number and complex number 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 denoting the height of a growing flower at time would be considered continuous. In contrast, the function denoting the amount of money in a bank account at time would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.

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History A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of $y\; =\; f(x)$ as follows: an infinitely small increment $\backslash alpha$ of the independent variable x always produces an infinitely small change $f(x+\backslash alpha)-f(x)$ of the dependent variable y (see e.g. Cours d'Analyse, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity). The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano, Karl Weierstrass denied continuity of a function at a point c unless it was defined at and on both sides of c, but Édouard Goursat allowed the function to be defined only at and on one side of c, and Camille Jordan allowed it even if the function was defined only at c. All three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.

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Real functions

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Definition A real function that is a function from 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.

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

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 $(-\backslash infty,\; +\backslash infty)$ (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 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 $f(x)\; =\; \backslash sqrt\{x\}$ is continuous on its whole domain, which is the closed interval $[0,+\backslash infty).$

Many commonly encountered functions are that have a domain formed by all real numbers, except some . Examples include the reciprocal function $x\; \backslash mapsto\; \backslash frac\; \{1\}\{x\}$ and the tangent function $x\backslash mapsto\; \backslash tan\; x.$ 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 $x\backslash mapsto\; \backslash frac\; \{1\}\{x\}$ and $x\backslash mapsto\; \backslash sin(\backslash frac\; \{1\}\{x\})$ are discontinuous at , and remain discontinuous whichever value is chosen for defining them at . 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 $f\; :\; D\; \backslash to\; \backslash R$ be a function whose domain $D$ is contained in $\backslash R$ of real numbers.

Some (but not all) possibilities for $D$ are:

• $D$ is the whole real line; that is, $D\; =\; \backslash R$
• $D$ is a closed interval of the form $D\; =\; a,\; =\; \backslash \{x\; \backslash in\; \backslash R\; \backslash mid\; a\; \backslash leq\; x\; \backslash leq\; b\; \backslash \}\; ,$ where and are real numbers
• $D$ is an open interval of the form where and are real numbers

In the case of an open interval, $a$ and $b$ do not belong to $D$, and the values $f(a)$ and $f(b)$ are not defined, and if they are, they do not matter for continuity on $D$.

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Definition in terms of limits of functions The function is continuous at some point of its domain if the limit of $f(x),$ as x approaches c through the domain of f, exists and is equal to $f(c).$, section II.4 In mathematical notation, this is written as $$\backslash lim\_\{x\; \backslash to\; c\}\{f(x)\}\; =\; f(c).$$ In detail this means three conditions: first, has to be defined at (guaranteed by the requirement that is in the domain of ). Second, the limit of that equation has to exist. Third, the value of this limit must equal $f(c).$

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

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Definition in terms of neighborhoods A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point $f(c)$ as the width of the neighborhood around c shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood $N\_1(f(c))$ there is a neighborhood $N\_2(c)$ in its domain such that $f(x)\; \backslash in\; N\_1(f(c))$ whenever $x\backslash in\; N\_2(c).$

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.

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Definition in terms of limits of sequences One can instead require that for any sequence $(x\_n)\_\{n\; \backslash in\; \backslash N\}$ of points in the domain which converges to c, the corresponding sequence $\backslash left(f(x\_n)\backslash right)\_\{n\backslash in\; \backslash N\}$ converges to $f(c).$ In mathematical notation, $$\backslash forall\; (x\_n)\_\{n\; \backslash in\; \backslash N\}\; \backslash subset\; D:\backslash lim\_\{n\backslash to\backslash infty\}\; x\_n\; =\; c\; \backslash Rightarrow\; \backslash lim\_\{n\backslash to\backslash infty\}\; f(x\_n)\; =\; f(c)\backslash ,.$$

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Weierstrass and Jordan definitions (epsilon–delta) of continuous functions Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function $f\; :\; D\; \backslash to\; \backslash mathbb\{R\}$ as above and an element $x\_0$ of the domain $D$, $f$ is said to be continuous at the point $x\_0$ when the following holds: For any positive real number $\backslash varepsilon\; >\; 0,$ however small, there exists some positive real number $\backslash delta\; >\; 0$ such that for all $x$ in the domain of $f$ with the value of $f(x)$ satisfies

Alternatively written, continuity of $f\; :\; D\; \backslash to\; \backslash mathbb\{R\}$ at $x\_0\; \backslash in\; D$ means that for every $\backslash varepsilon\; >\; 0,$ there exists a $\backslash delta\; >\; 0$ such that for all $x\; \backslash in\; D$:

More intuitively, we can say that if we want to get all the $f(x)$ values to stay in some small neighborhood around $f\backslash left(x\_0\backslash right),$ we need to choose a small enough neighborhood for the $x$ values around $x\_0.$ If we can do that no matter how small the $f(x\_0)$ neighborhood is, then $f$ is continuous at $x\_0.$

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

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

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Definition in terms of control of the remainder In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity. A function $C:\; 0,\backslash infty)$ is called a control function if

• C is non-decreasing
• $\backslash inf\_\{\backslash delta\; >\; 0\}\; C(\backslash delta)\; =\; 0$

A function $f\; :\; D\; \backslash to\; R$ is C-continuous at $x\_0$ if there exists such a neighbourhood $N(x\_0)$ that $$|f(x)\; -\; f(x\_0)|\; \backslash leq\; C\backslash left(\backslash left|x\; -\; x\_0\backslash right|\backslash right)\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; D\; \backslash cap\; N(x\_0)$$

A function is continuous in $x\_0$ if it is C-continuous for some control function C.

This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions $\backslash mathcal\{C\}$ a function is if it is for some $C\; \backslash in\; \backslash mathcal\{C\}.$ For example, the Lipschitz, the Hölder continuous functions of exponent and the uniformly continuous functions below are defined by the set of control functions $$\backslash mathcal\{C\}\_\{\backslash mathrm\{Lipschitz\}\}\; =\; \backslash \{C\; :\; C(\backslash delta)\; =\; K|\backslash delta|\; ,\backslash \; K\; >\; 0\backslash \}$$ $$\backslash mathcal\{C\}\_\{\backslash text\{Hölder\}-\backslash alpha\}\; =\; \backslash \{C\; :\; C(\backslash delta)\; =\; K\; |\backslash delta|^\backslash alpha,\; \backslash \; K\; >\; 0\backslash \}$$ $$\backslash mathcal\{C\}\_\{\backslash text\{uniform\; cont.\}\}\; =\; \backslash \{C\; :\; C(0)\; =\; 0\; \backslash \}$$ respectively.

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Definition using oscillation Continuity can also be defined in terms of oscillation: a function f is continuous at a point $x\_0$ if and only if its oscillation at that point is zero; Introduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172 in symbols, $\backslash omega\_f(x\_0)\; =\; 0.$ A benefit of this definition is that it discontinuity: the oscillation gives how 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 $\backslash varepsilon$ (hence a G-delta set) – and gives a rapid proof of one direction of the Lebesgue integrability condition. 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. 171–177

The oscillation is equivalent to the $\backslash varepsilon-\backslash delta$ 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 $\backslash varepsilon\_0$ there is no $\backslash delta$ that satisfies the $\backslash varepsilon-\backslash delta$ definition, then the oscillation is at least $\backslash varepsilon\_0,$ and conversely if for every $\backslash varepsilon$ there is a desired $\backslash delta,$ the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.

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Definition using the hyperreals Cauchy defined the continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34). Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows.

(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.

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Rules for continuity Proving the continuity of a function by a direct application of the definition is generaly a noneasy task. Fortunately, in practice, most functions are built from simpler functions, and their continuity can be deduced immediately from the way they are defined, by applying the following rules:

• Every constant function is continuous
• The identity function is continuous
• Addition and multiplication: If the functions and are continuous on their respective domains and , then their sum and their product are continuous on the set intersection , where and are defined by and .
• Reciprocal: If the function is continuous on the domain , then its reciprocal , defined by is continuous on the domain , that is, the domain from which the points such that are removed.
• Function composition: If the functions and are continuous on their respective domains and , then the composition defined by is continuous on , that the part of that is mapped by inside .
• The sine and cosine functions ( and ) are continuous everywhere.
• The exponential function is continuous everywhere.
• The natural logarithm is continuous on the domain formed by all positive real numbers .

These rules imply that every polynomial function is continuous everywhere and that a rational function is continuous everywhere where it is defined, if the numerator and the denominator have no common zeros. More generally, the quotient of two continuous functions is continuous outside the zeros of the denominator.

An example of a function for which the above rules are not sufficirent is the sinc function, which is defined by and for . The above rules show immediately that the function is continuous for , but, for proving the continuity at , one has to prove $$\backslash lim\_\{x\backslash to\; 0\}\; \backslash frac\{\backslash sin\; x\}\{x\}\; =\; 1.$$ As this is true, one gets that the sinc function is continuous function on all real numbers.

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Examples of discontinuous functions An example of a discontinuous function is the Heaviside step function $H$, defined by

Pick for instance $\backslash varepsilon\; =\; 1/2$. Then there is no around $x\; =\; 0$, i.e. no open interval $(-\backslash delta,\backslash ;\backslash delta)$ with $\backslash delta\; >\; 0,$ that will force all the $H(x)$ values to be within the of $H(0)$, i.e. within $(1/2,\backslash ;3/2)$. Intuitively, we can think of this type of discontinuity as a sudden jump in function values.

Similarly, the Sign function or sign function is discontinuous at $x\; =\; 0$ but continuous everywhere else. Yet another example: the function $$f(x)\; =\; \backslash begin\{cases\}$$

 \sin\left(x^{-2}\right)&\text{ if }x \neq 0\\
  0&\text{ if }x = 0
     

\end{cases} is continuous everywhere apart from $x\; =\; 0$.

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

 0&\text{ if }x\text{ is irrational}.
     

\end{cases} 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, $$D(x)=\backslash begin\{cases\}$$

 0&\text{ if }x\text{  is irrational } (\in \R \setminus \Q)\\
 1&\text{ if }x\text{ is rational } (\in \Q)
     

\end{cases} is nowhere continuous.

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Properties

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A useful lemma Let $f(x)$ be a function that is continuous at a point $x\_0,$ and $y\_0$ be a value such $f\backslash left(x\_0\backslash right)\backslash neq\; y\_0.$ Then $f(x)\backslash neq\; y\_0$ throughout some neighbourhood of $x\_0.$

Proof: By the definition of continuity, take $\backslash varepsilon\; =\backslash frac$

{2}>0 , then there exists $\backslash delta>0$ such that Suppose there is a point in the neighbourhood for which $f(x)=y\_0;$ then we have the contradiction

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

If the real-valued function f is continuous on the closed interval $a,,$ and k is some number between $f(a)$ and $f(b),$ then there is some number $c\; \backslash in\; a,,$ such that $f(c)\; =\; k.$

For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.

As a consequence, if f is continuous on $a,$ and $f(a)$ and $f(b)$ differ in sign, then, at some point $c\; \backslash in\; a,,$ $f(c)$ must equal zero.

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Extreme value theorem The extreme value theorem states that if a function f is defined on a closed interval $a,$ (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists $c\; \backslash in\; a,$ with $f(c)\; \backslash geq\; f(x)$ for all $x\; \backslash in\; a,.$ 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 $f(x)\; =\; \backslash frac\{1\}\{x\},$ defined on the open interval (0,1), does not attain a maximum, being unbounded above.

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Relation to differentiability and integrability Every differentiable function $$f\; :\; (a,\; b)\; \backslash to\; \backslash R$$ is continuous, as can be shown. The converse does not hold: for example, the absolute value function

$f(x)=|x|\; =\; \backslash begin\{cases\}$

 \;\;\ x & \text{ if }x \geq 0\\
 -x & \text{ if }x < 0
     

\end{cases} is everywhere continuous. However, it is not differentiable at $x\; =\; 0$ (but is so everywhere else). Weierstrass's function is also everywhere continuous but nowhere differentiable.

The derivative f′( x) of a differentiable function f( x) need not be continuous. If f′( x) is continuous, f( x) is said to be continuously differentiable. The set of such functions is denoted $C^1((a,\; b)).$ More generally, the set of functions $$f\; :\; \backslash Omega\; \backslash to\; \backslash R$$ (from an open interval (or open subset of $\backslash R$) $\backslash Omega$ to the reals) such that f is $n$ times differentiable and such that the $n$-th derivative of f is continuous is denoted $C^n(\backslash Omega).$ See differentiability class. In the field of computer graphics, properties related (but not identical) to $C^0,\; C^1,\; C^2$ are sometimes called $G^0$ (continuity of position), $G^1$ (continuity of tangency), and $G^2$ (continuity of curvature); see Smoothness of curves and surfaces.

Every continuous function $$f\; :\; a,\; \backslash to\; \backslash R$$ is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the (integrable but discontinuous) sign function shows.

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Pointwise and uniform limits Given a sequence $$f\_1,\; f\_2,\; \backslash dotsc\; :\; I\; \backslash to\; \backslash R$$ of functions such that the limit $$f(x)\; :=\; \backslash lim\_\{n\; \backslash to\; \backslash infty\}\; f\_n(x)$$ exists for all $x\; \backslash in\; D,$, the resulting function $f(x)$ is referred to as the pointwise limit of the sequence of functions $\backslash left(f\_n\backslash right)\_\{n\; \backslash in\; N\}.$ The pointwise limit function need not be continuous, even if all functions $f\_n$ are continuous, as the animation at the right shows. However, f is continuous if all functions $f\_n$ are continuous and the sequence converges uniformly, by the uniform convergence theorem. This theorem can be used to show that the exponential functions, , square root function, and trigonometric functions are continuous.

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Directional Continuity Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi-continuity. Roughly speaking, a function is 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 $\backslash varepsilon\; >\; 0$ however small, there exists some number $\backslash delta\; >\; 0$ such that for all x in the domain with the value of $f(x)$ 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 functions. A function is continuous if and only if it is both right-continuous and left-continuous.

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Semicontinuity A function f is if, roughly, any jumps that might occur only go down, but not up. That is, for any $\backslash varepsilon\; >\; 0,$ there exists some number $\backslash delta\; >\; 0$ such that for all x in the domain with the value of $f(x)$ satisfies $$f(x)\; \backslash geq\; f(c)\; -\; \backslash epsilon.$$ The reverse condition is .

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Continuous functions between metric spaces The concept of continuous real-valued functions can be generalized to functions between . A metric space is a set $X$ equipped with a function (called metric) $d\_X,$ that can be thought of as a measurement of the distance of any two elements in X. Formally, the metric is a function $$d\_X\; :\; X\; \backslash times\; X\; \backslash to\; \backslash R$$ that satisfies a number of requirements, notably the triangle inequality. Given two metric spaces $\backslash left(X,\; d\_X\backslash right)$ and $\backslash left(Y,\; d\_Y\backslash right)$ and a function $$f\; :\; X\; \backslash to\; Y$$ then $f$ is continuous at the point $c\; \backslash in\; X$ (with respect to the given metrics) if for any positive real number $\backslash varepsilon\; >\; 0,$ there exists a positive real number $\backslash delta\; >\; 0$ such that all $x\; \backslash in\; X$ satisfying will also satisfy As in the case of real functions above, this is equivalent to the condition that for every sequence $\backslash left(x\_n\backslash right)$ in $X$ with limit $\backslash lim\; x\_n\; =\; c,$ we have $\backslash lim\; f\backslash left(x\_n\backslash right)\; =\; f(c).$ The latter condition can be weakened as follows: $f$ is continuous at the point $c$ if and only if for every convergent sequence $\backslash left(x\_n\backslash right)$ in $X$ with limit $c$, the sequence $\backslash left(f\backslash left(x\_n\backslash right)\backslash right)$ is a Cauchy sequence, and $c$ is in the domain of $f$.

The set of points at which a function between metric spaces is continuous is a $G\_\{\backslash delta\}$ set – this follows from the $\backslash varepsilon-\backslash delta$ definition of continuity.

This notion of continuity is applied, for example, in functional analysis. A key statement in this area says that a linear operator $$T\; :\; V\; \backslash to\; W$$ between normed vector spaces $V$ and $W$ (which are equipped with a compatible norm, denoted $\backslash |x\backslash |$) is continuous if and only if it is bounded, that is, there is a constant $K$ such that $$\backslash |T(x)\backslash |\; \backslash leq\; K\; \backslash |x\backslash |$$ for all $x\; \backslash in\; V.$

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Uniform, Hölder and Lipschitz continuity The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way $\backslash delta$ depends on $\backslash varepsilon$ and c in the definition above. Intuitively, a function f as above is uniformly continuous if the $\backslash delta$ does not depend on the point c. More precisely, it is required that for every real number $\backslash varepsilon\; >\; 0$ there exists $\backslash delta\; >\; 0$ such that for every $c,\; b\; \backslash in\; X$ 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 ., section IV.10

A function is Hölder continuous with exponent α (a real number) if there is a constant K such that for all $b,\; c\; \backslash in\; X,$ the inequality $$d\_Y\; (f(b),\; f(c))\; \backslash leq\; K\; \backslash cdot\; (d\_X\; (b,\; c))^\backslash alpha$$ holds. Any Hölder continuous function is uniformly continuous. The particular case $\backslash alpha\; =\; 1$ is referred to as Lipschitz continuity. That is, a function is Lipschitz continuous if there is a constant K such that the inequality $$d\_Y\; (f(b),\; f(c))\; \backslash leq\; K\; \backslash cdot\; d\_X\; (b,\; c)$$ holds for any $b,\; c\; \backslash in\; X.$, section 9.4 The Lipschitz condition occurs, for example, in the Picard–Lindelöf theorem concerning the solutions of ordinary differential equations.

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Continuous functions between topological spaces Another, more abstract, notion of continuity is the continuity of functions between topological spaces in which there generally is no formal notion of distance, as there is in the case of . A topological space is a set X together with a topology on X, which is a set of of X satisfying a few requirements with respect to their unions and intersections that generalize the properties of the in metric spaces while still allowing one to talk about the neighborhoods of a given point. The elements of a topology are called of X (with respect to the topology).

A function $$f\; :\; X\; \backslash to\; Y$$ between two topological spaces X and Y is continuous if for every open set $V\; \backslash subseteq\; Y,$ the inverse image $$f^\{-1\}(V)\; =\; \backslash \{x\; \backslash in\; X\; \backslash ;\; |\; \backslash ;\; f(x)\; \backslash in\; V\; \backslash \}$$ 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 $T\_X$), 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 (which are the complements of the open subsets) in Y are closed in X.

An extreme example: if a set X is given the discrete topology (in which every subset is open), all functions $$f\; :\; X\; \backslash to\; T$$ to any topological space T are continuous. On the other hand, if X is equipped with the indiscrete topology (in which the only open subsets are the empty set and X) and the space T set is at least T₀, then the only continuous functions are the constant functions. Conversely, any function whose codomain is indiscrete is continuous.

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Continuity at a point The translation in the language of neighborhoods of the $(\backslash varepsilon,\; \backslash delta)$-definition of continuity leads to the following definition of the continuity at a point:

This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using rather than images.

Also, as every set that contains a neighborhood is also a neighborhood, and $f^\{-1\}(V)$ is the largest subset of such that $f(U)\; \backslash subseteq\; V,$ this definition may be simplified into:

As an open set is a set that is a neighborhood of all its points, a function $f\; :\; X\; \backslash to\; Y$ is continuous at every point of if and only if it is a continuous function.

If X and Y are metric spaces, it is equivalent to consider the neighborhood system of centered at x and f( x) instead of all neighborhoods. This gives back the above $\backslash varepsilon-\backslash delta$ 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 $x\; \backslash in\; X,$ a map $f\; :\; X\; \backslash to\; Y$ is continuous at $x$ if and only if whenever $\backslash mathcal\{B\}$ is a filter on $X$ that converges to $x$ in $X,$ which is expressed by writing $\backslash mathcal\{B\}\; \backslash to\; x,$ then necessarily $f(\backslash mathcal\{B\})\; \backslash to\; f(x)$ in $Y.$ If $\backslash mathcal\{N\}(x)$ denotes the neighborhood filter at $x$ then $f\; :\; X\; \backslash to\; Y$ is continuous at $x$ if and only if $f(\backslash mathcal\{N\}(x))\; \backslash to\; f(x)$ in $Y.$ Moreover, this happens if and only if the prefilter $f(\backslash mathcal\{N\}(x))$ is a filter base for the neighborhood filter of $f(x)$ in $Y.$

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

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Sequences and nets In several contexts, the topology of a space is conveniently specified in terms of limit points. This is often accomplished by specifying when a point is the limit of a sequence. Still, for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points Indexed family 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 $f\; :\; X\; \backslash to\; Y$ is sequentially continuous if whenever a sequence $\backslash left(x\_n\backslash right)$ in $X$ converges to a limit $x,$ the sequence $\backslash left(f\backslash left(x\_n\backslash right)\backslash right)$ converges to $f(x).$ Thus, sequentially continuous functions "preserve sequential limits." Every continuous function is sequentially continuous. If $X$ is a first-countable space and countable choice holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if $X$ is a metric space, sequential continuity and continuity are equivalent. For non-first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called .) 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:

Proof. Assume that $f\; :\; A\; \backslash subseteq\; \backslash R\; \backslash to\; \backslash R$ is continuous at $x\_0$ (in the sense of $\backslash epsilon-\backslash delta$ continuity). Let $\backslash left(x\_n\backslash right)\_\{n\backslash geq1\}$ be a sequence converging at $x\_0$ (such a sequence always exists, for example, $x\_n\; =\; x,\; \backslash text\{\; for\; all\; \}\; n$); since $f$ is continuous at $x\_0$ For any such $\backslash delta\_\{\backslash epsilon\}$ we can find a natural number $\backslash nu\_\{\backslash epsilon\}\; >\; 0$ such that for all $n\; >\; \backslash nu\_\{\backslash epsilon\},$ since $\backslash left(x\_n\backslash right)$ converges at $x\_0$; combining this with $(*)$ we obtain Assume on the contrary that $f$ is sequentially continuous and proceed by contradiction: suppose $f$ is not continuous at $x\_0$ then we can take $\backslash delta\_\{\backslash epsilon\}=1/n,\backslash ,\backslash forall\; n\; >\; 0$ and call the corresponding point $x\_\{\backslash delta\_\{\backslash epsilon\}\}\; =:\; x\_n$: in this way we have defined a sequence $(x\_n)\_\{n\backslash geq1\}$ such that by construction $x\_n\; \backslash to\; x\_0$ but $f(x\_n)\; \backslash not\backslash to\; f(x\_0)$, which contradicts the hypothesis of sequential continuity. $\backslash blacksquare$

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Closure operator and interior operator definitions In terms of the interior and closure operators, we have the following equivalences,

Proof. i ⇒ ii. Fix a subset $B$ of $Y.$ Since $\backslash operatorname\{int\}\_Y\; B$ is open. and $f$ is continuous, $f^\{-1\}(\backslash operatorname\{int\}\_Y\; B)$ is open in $X.$ As $\backslash operatorname\{int\}\_Y\; B\; \backslash subseteq\; B,$ we have $f^\{-1\}(\backslash operatorname\{int\}\_Y\; B)\; \backslash subseteq\; f^\{-1\}(B).$ By the definition of the interior, $\backslash operatorname\{int\}\_X\backslash left(f^\{-1\}(B)\backslash right)$ is the largest open set contained in $f^\{-1\}(B).$ Hence $f^\{-1\}(\backslash operatorname\{int\}\_Y\; B)\; \backslash subseteq\; \backslash operatorname\{int\}\_X\backslash left(f^\{-1\}(B)\backslash right).$

ii ⇒ iii. Fix $A\backslash subseteq\; X$ and let $x\backslash in\backslash operatorname\{cl\}\_X\; A.$ Suppose to the contrary that $f(x)\backslash notin\backslash operatorname\{cl\}\_Y\backslash left(f(A)\backslash right),$ then we may find some open neighbourhood $V$ of $f(x)$ that is disjoint from $\backslash operatorname\{cl\}\_Y\backslash left(f(A)\backslash right)$. By ii, $f^\{-1\}(V)\; =\; f^\{-1\}(\backslash operatorname\{int\}\_Y\; V)\; \backslash subseteq\; \backslash operatorname\{int\}\_X\; \backslash left(f^\{-1\}(V)\backslash right),$ hence $f^\{-1\}(V)$ is open. Then we have found an open neighbourhood of $x$ that does not intersect $\backslash operatorname\{cl\}\_X\; A$, contradicting the fact that $x\backslash in\backslash operatorname\{cl\}\_X\; A.$ Hence $f\backslash left(\backslash operatorname\{cl\}\_X\; A\backslash right)\; \backslash subseteq\; \backslash operatorname\{cl\}\_Y\; \backslash left(f(A)\backslash right).$

iii ⇒ i. Let $N\backslash subseteq\; Y$ be closed. Let $M\; =\; f^\{-1\}(N)$ be the preimage of $N.$ By iii, we have $f\backslash left(\backslash operatorname\{cl\}\_X\; M\backslash right)\; \backslash subseteq\; \backslash operatorname\{cl\}\_Y\; \backslash left(f(M)\backslash right).$ Since $f(M)\; =\; f(f^\{-1\}(N))\; \backslash subseteq\; N,$ we have further that $f\backslash left(\backslash operatorname\{cl\}\_X\; M\backslash right)\; \backslash subseteq\; \backslash operatorname\{cl\}\_Y\; N\; =\; N.$ Thus $\backslash operatorname\{cl\}\_X\; M\; \backslash subseteq\; f^\{-1\}\backslash left(f(\backslash operatorname\{cl\}\_X\; M)\backslash right)\; \backslash subseteq\; f^\{-1\}(N)\; =\; M.$ Hence $M$ is closed and we are done.

If we declare that a point $x$ is a subset $A\; \backslash subseteq\; X$ if $x\; \backslash in\; \backslash operatorname\{cl\}\_X\; A,$ then this terminology allows for a plain English description of continuity: $f$ is continuous if and only if for every subset $A\; \backslash subseteq\; X,$ $f$ maps points that are close to $A$ to points that are close to $f(A).$ Similarly, $f$ is continuous at a fixed given point $x\; \backslash in\; X$ if and only if whenever $x$ is close to a subset $A\; \backslash subseteq\; X,$ then $f(x)$ is close to $f(A).$

Instead of specifying topological spaces by their Open set, any topology on $X$ can alternatively be determined by a closure operator or by an interior operator. Specifically, the map that sends a subset $A$ of a topological space $X$ to its topological closure $\backslash operatorname\{cl\}\_X\; A$ satisfies the Kuratowski closure axioms. Conversely, for any closure operator $A\; \backslash mapsto\; \backslash operatorname\{cl\}\; A$ there exists a unique topology $\backslash tau$ on $X$ (specifically, $\backslash tau\; :=\; \backslash \{\; X\; \backslash setminus\; \backslash operatorname\{cl\}\; A\; :\; A\; \backslash subseteq\; X\; \backslash \}$) such that for every subset $A\; \backslash subseteq\; X,$ $\backslash operatorname\{cl\}\; A$ is equal to the topological closure $\backslash operatorname\{cl\}\_\{(X,\; \backslash tau)\}\; A$ of $A$ in $(X,\; \backslash tau).$ If the sets $X$ and $Y$ are each associated with closure operators (both denoted by $\backslash operatorname\{cl\}$) then a map $f\; :\; X\; \backslash to\; Y$ is continuous if and only if $f(\backslash operatorname\{cl\}\; A)\; \backslash subseteq\; \backslash operatorname\{cl\}\; (f(A))$ for every subset $A\; \backslash subseteq\; X.$

Similarly, the map that sends a subset $A$ of $X$ to its topological interior $\backslash operatorname\{int\}\_X\; A$ defines an interior operator. Conversely, any interior operator $A\; \backslash mapsto\; \backslash operatorname\{int\}\; A$ induces a unique topology $\backslash tau$ on $X$ (specifically, $\backslash tau\; :=\; \backslash \{\; \backslash operatorname\{int\}\; A\; :\; A\; \backslash subseteq\; X\; \backslash \}$) such that for every $A\; \backslash subseteq\; X,$ $\backslash operatorname\{int\}\; A$ is equal to the topological interior $\backslash operatorname\{int\}\_\{(X,\; \backslash tau)\}\; A$ of $A$ in $(X,\; \backslash tau).$ If the sets $X$ and $Y$ are each associated with interior operators (both denoted by $\backslash operatorname\{int\}$) then a map $f\; :\; X\; \backslash to\; Y$ is continuous if and only if $f^\{-1\}(\backslash operatorname\{int\}\; B)\; \backslash subseteq\; \backslash operatorname\{int\}\backslash left(f^\{-1\}(B)\backslash right)$ for every subset $B\; \backslash subseteq\; Y.$

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

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Properties If $f\; :\; X\; \backslash to\; Y$ and $g\; :\; Y\; \backslash to\; Z$ are continuous, then so is the composition $g\; \backslash circ\; f\; :\; X\; \backslash to\; Z.$ If $f\; :\; X\; \backslash to\; Y$ is continuous and

• X is Compact space, then f( X) is compact.
• X is Connected space, then f( X) is connected.
• X is path-connected, then f( X) is path-connected.
• X is Lindelöf, then f( X) is Lindelöf.
• X is separable space, then f( X) is separable.

The possible topologies on a fixed set X are partial ordering: a topology $\backslash tau\_1$ is said to be coarser than another topology $\backslash tau\_2$ (notation: $\backslash tau\_1\; \backslash subseteq\; \backslash tau\_2$) if every open subset with respect to $\backslash tau\_1$ is also open with respect to $\backslash tau\_2.$ Then, the identity map $$\backslash operatorname\{id\}\_X\; :\; \backslash left(X,\; \backslash tau\_2\backslash right)\; \backslash to\; \backslash left(X,\; \backslash tau\_1\backslash right)$$ is continuous if and only if $\backslash tau\_1\; \backslash subseteq\; \backslash tau\_2$ (see also comparison of topologies). More generally, a continuous function $$\backslash left(X,\; \backslash tau\_X\backslash right)\; \backslash to\; \backslash left(Y,\; \backslash tau\_Y\backslash right)$$ stays continuous if the topology $\backslash tau\_Y$ is replaced by a coarser topology and/or $\backslash tau\_X$ is replaced by a finer topology.

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Homeomorphisms Symmetric to the concept of a continuous map is an open map, for which of open sets are open. If an open map f has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a bijective function f between two topological spaces, the inverse function $f^\{-1\}$ need not be continuous. A bijective continuous function with a continuous inverse function is called a .

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

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Defining topologies via continuous functions Given a function $$f\; :\; X\; \backslash to\; S,$$ where X is a topological space and S is a set (without a specified topology), the final topology on S is defined by letting the open sets of S be those subsets A of S for which $f^\{-1\}(A)$ is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on S. Thus, the final topology is the finest topology on S that makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f.

Dually, for a function f from a set S to a topological space X, the initial topology on S is defined by designating as an open set every subset A of S such that $A\; =\; f^\{-1\}(U)$ for some open subset U of X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus, the initial topology is the coarsest topology on S that makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X.

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

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Related notions If $f\; :\; S\; \backslash to\; Y$ is a continuous function from some subset $S$ of a topological space $X$ then a of $f$ to $X$ is any continuous function $F\; :\; X\; \backslash to\; Y$ such that $F(s)\; =\; f(s)$ for every $s\; \backslash in\; S,$ which is a condition that often written as $f\; =\; F\backslash big\backslash vert\_S.$ In words, it is any continuous function $F\; :\; X\; \backslash to\; Y$ that restricts to $f$ on $S.$ This notion is used, for example, in the Tietze extension theorem and the Hahn–Banach theorem. If $f\; :\; S\; \backslash to\; Y$ is not continuous, then it could not possibly have a continuous extension. If $Y$ is a Hausdorff space and $S$ is a Dense set of $X$ then a continuous extension of $f\; :\; S\; \backslash to\; Y$ to $X,$ if one exists, will be unique. The Blumberg theorem states that if $f\; :\; \backslash R\; \backslash to\; \backslash R$ is an arbitrary function then there exists a dense subset $D$ of $\backslash R$ such that the restriction $f\backslash big\backslash vert\_D\; :\; D\; \backslash to\; \backslash R$ is continuous; in other words, every function $\backslash R\; \backslash to\; \backslash R$ can be restricted to some dense subset on which it is continuous.

Various other mathematical domains use the concept of continuity in different but related meanings. For example, in order theory, an order-preserving function $f\; :\; X\; \backslash to\; Y$ between particular types of partially ordered sets $X$ and $Y$ is continuous if for each Directed set $A$ of $X,$ we have $\backslash sup\; f(A)\; =\; f(\backslash sup\; A).$ Here $\backslash ,\backslash sup\backslash ,$ is the supremum with respect to the orderings in $X$ and $Y,$ respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the Scott topology.

(2025). 9780521803380, Cambridge University Press. . ISBN 9780521803380

In category theory, a functor $$F\; :\; \backslash mathcal\; C\; \backslash to\; \backslash mathcal\; D$$ between two categories is called if it commutes with small limits. That is to say, $$\backslash varprojlim\_\{i\; \backslash in\; I\}\; F(C\_i)\; \backslash cong\; F\; \backslash left(\backslash varprojlim\_\{i\; \backslash in\; I\}\; C\_i\; \backslash right)$$ for any small (that is, indexed by a set $I,$ as opposed to a class) diagram of objects in $\backslash mathcal\; C$.

A 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 Domain theory.

In measure theory, a function $f\; :\; E\; \backslash to\; \backslash mathbb\{R\}^k$ defined on a Lebesgue measurable set $E\; \backslash subseteq\; \backslash mathbb\{R\}^n$ is called approximately continuous at a point $x\_0\; \backslash in\; E$ if the approximate limit of $f$ at $x\_0$ exists and equals $f(x\_0)$. This generalizes the notion of continuity by replacing the ordinary limit with the approximate limit. A fundamental result known as the Stepanov-Denjoy theorem states that a function is measurable if and only if it is approximately continuous almost everywhere.

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See also

• Continuity (mathematics)
• Absolute continuity
• Approximate continuity
• Dini continuity
• Equicontinuity
• Geometric continuity
• Parametric continuity
• Classification of discontinuities
• Coarse function
• Continuous function (set theory)
• Continuous stochastic process
• Normal function
• Open and closed maps
• Piecewise
• Symmetrically continuous function

• Direction-preserving function - an analog of a continuous function in discrete spaces.

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Bibliography

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