In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics, matrix calculus is a specialized notation for doing Multivariable calculus, especially over spaces of matrices, where it defines the In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable In Calculus, the quotient rule is a method of finding the Derivative of a function that is the Quotient of two other functions for which In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. In Mathematics, an implicit function is a generalization for the concept of a function in which the Dependent variable has not been given "explicitly" In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor In Differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change The primary operation in Differential calculus is finding a Derivative. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space See the following pages for lists of Integrals: List of integrals of rational functions List of integrals of irrational functions In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other Disk integration is a means of calculating the Volume of a Solid of revolution, when integrating along the axis of revolution Shell integration (the shell method in Integral calculus) is a means of calculating the Volume of a Solid of revolution, when integrating In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions In Integral calculus, the use of Partial fractions is required to integrate the general Rational function. In Calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other hopefully simpler integrals by Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Otherwise, a function is said to be discontinuous. A continuous function with a continuous inverse function is called bicontinuous. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B An intuitive though imprecise (and inexact) idea of continuity is given by the common statement that a continuous function is a function whose graph can be drawn without lifting the chalk from the blackboard.

Continuity of functions is one of the core concepts of topology, which is treated in full generality in a more advanced article. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function This introductory article focuses on the special case where the inputs and outputs of functions are real numbers. In Mathematics, the real numbers may be described informally in several different ways In addition, this article discusses the definition for the more general case of functions between two metric spaces. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying Domain theory is a branch of Mathematics that studies special kinds of Partially ordered sets (posets commonly called domains. In Mathematics, a function between two Partially ordered sets P and Q is Scott-continuous (named after the mathematician Dana

As an example, consider the function h(t) which describes the height of a growing flower at time t. Height is the measurement of vertical Distance, but has two meanings in common use This function is continuous. In fact, there is a dictum of classical physics which states that in nature everything is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.

Real-valued continuous functions

Suppose we have a function that maps real numbers to real numbers and whose domain is some interval, like the functions h and M above. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps". In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object

To be more precise, we say that the function f is continuous at some point c when the following two requirements are satisfied:

• f(c) must be defined (i. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume e. c must be an element of the domain of f). In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined
• The limit of f(x) as x approaches c must exist and be equal to f(c). In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" (If the point c in the domain of f is not a limit point of the domain, then this condition is vacuously true, since x cannot approach c. In Mathematics, informally speaking a limit point of a set S in a Topological space X is a point x in X that can be "approximated" A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If  A  then  Thus, for example, every function whose domain is the set of all integers is continuous, merely for lack of opportunity to be otherwise. However, one does not usually talk about continuous functions in this setting. )

We call the function everywhere continuous, or simply continuous, if it is continuous at every point of its domain. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined More generally, we say that a function is continuous on some subset of its domain if it is continuous at every point of that subset. If we simply say that a function is continuous, we usually mean that it is continuous for all real numbers.

The notation C(Ω) or C⁰(Ω) is sometimes used to denote the set of all continuous functions with domain Ω. Similarly, C¹(Ω) is used to denote the set of differentiable functions whose derivative is continuous, C²(Ω) for the twice-differentiable functions whose second derivative is continuous, and so on. In the field of computer graphics, these three levels are sometimes called g0 (continuity of position), g1 (continuity of tangency), and g2 (continuity of curvature). The notation C^{(n, α)}(Ω) occurs in the definition of a more subtle concept, that of Hölder continuity. In Mathematics, a real-valued function f on R n satisfies a Hölder condition, or is Hölder continuous, when there are

Cauchy definition (epsilon-delta)

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

Again consider a function f that maps a set of real numbers to another set of real numbers, and suppose c is an element of the domain of f. In Mathematics, the real numbers may be described informally in several different ways The function f is said to be continuous at the point c if the following holds: For any number ε > 0, however small, there exists some number δ > 0 such that for all x in the domain with c − δ < x < c + δ, the value of f(x) satisfies

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

This "epsilon-delta definition" of continuity was first given by Cauchy.

More intuitively, we can say that if we want to get all the f(x) values to stay in some small neighborhood around f(c), we simply need to choose a small enough neighborhood for the x values around c, and we can do that no matter how small the f(x) neighborhood is; f is then continuous at c. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space.

Heine definition of continuity

The following definition of continuity is due to Heine. Heinrich Eduard Heine ( March 15 1821 &ndash October 21, 1881) was a German mathematician.

A real function f is continuous if for any sequence (x_n) such that

it holds that

(We assume that all points x_n, x₀ belong to the domain of f. In Mathematics, a sequence is an ordered list of objects (or events )

One can say, briefly, that a function is continuous if and only if it preserves limits.

Cauchy's and Heine's definitions of continuity are equivalent on the reals. The usual (easier) proof makes use of the axiom of choice, but in the case of global continuity of real functions it was proved by Wacław Sierpiński that the axiom of choice is not actually needed. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. Wacław Franciszek Sierpiński ( March 14 1882 — October 21 1969) (ˈvaʦwaf fraɲˈʨiʂɛk ɕɛrˈpʲiɲskʲi a Polish Mathematician ^[1]

In more general setting of topological spaces, the concept analogous to Heine definition of continuity is called sequential continuity. In general, the condition of sequential continuity is weaker than the analogue of Cauchy continuity, which is just called continuity (see continuity (topology) for details). The Language of mathematics has a vast Vocabulary of specialist and technical terms In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function

Examples

• All polynomial functions are continuous. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations
• If a function has a domain which is not an interval, the notion of a continuous function as one whose graph you can draw without taking your pencil off the paper is not quite correct. Consider the functions f(x) = 1/x and g(x) = (sin x)/x. Neither function is defined at x = 0, so each has domain R \ {0} of real numbers except 0, and each function is continuous. In Mathematics, the real numbers may be described informally in several different ways The question of continuity at x = 0 does not arise, since it is not in the domain. The function f cannot be extended to a continuous function whose domain is R, since no matter what value is assigned at 0, the resulting function will not be continuous. On the other hand, since the limit of g at 0 is 1, g can be extended continuously to R by defining its value at 0 to be 1.
• The rational functions, exponential functions, logarithms, square root function, trigonometric functions and absolute value function are continuous. In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.
• An example of a discontinuous function is the function f defined by f(x) = 1 if x > 0, f(x) = 0 if x ≤ 0. Pick for instance ε = ½. There is no δ-neighborhood around x = 0 that will force all the f(x) values to be within ε of f(0). Intuitively we can think of this type of discontinuity as a sudden jump in function values.
• Another example of a discontinuous function is the sign function.
• A more complicated example of a discontinuous function is Thomae's function. If you are having difficulty understanding this article you might wish to learn more about Algebra, functions and mathematical limits.
• The function

is continuous at only one point, namely x = 0.

Facts about continuous functions

If two functions f and g are continuous, then f + g and fg are continuous. If g(x) ≠ 0 for all x in the domain, then f/g is also continuous.

The composition f o g of two continuous functions is continuous. In Mathematics, a composite function represents the application of one function to the results of another

If a function is differentiable at some point c of its domain, then it is also continuous at c. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The converse is not true: a function that's continuous at c need not be differentiable there. Consider for instance the absolute value function at c = 0. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

Intermediate value theorem

A continuous function on a closed interval has a maximum (green here) and a minimum (blue), and assumes all values between them.

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

If the real-valued function f is continuous on the closed interval [a, b] and k is some number between f(a) and f(b), then there is some number c in [a, b] such that f(c) = k. In Mathematical analysis, the intermediate value theorem states that for each value between the upper and lower bounds of the image of a Continuous function In Mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s. In Mathematics, the real numbers may be described informally in several different ways In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set

For example, if a child grows from 1m to 1. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International 5m between the ages of 2 years and 6 years, then, at some time between 2 years and 6 years of age, the child's height must have been 1. 25m.

As a consequence, if f is continuous on [a, b] and f(a) and f(b) differ in sign, then, at some point c in [a, b], f(c) must equal zero. A negative number is a Number that is less than zero, such as −2

Extreme value theorem

The extreme value theorem states that if a function f is defined on a closed interval [a,b] (or any closed and bounded set) and is continuous there, then the function attains its maximum, i. In Calculus, the extreme value theorem states that if a real-valued function f is continuous in the closed interval, then f e. there exists c ∈ [a,b] with f(c) ≥ f(x) for all x ∈ [a,b]. The same is true of the minimum of f. These statements are not, in general, true if the function is defined on an open interval (a,b) (or any set that is not both closed and bounded), as, for example, the continuous function f(x) = 1/x, defined on the open interval (0,1), does not attain a maximum, being unbounded above.

Directional continuity

A right continuous function

A left continuous function

A function may happen to be continuous in only one direction, either from the "left" or from the "right". A right-continuous function is a function which is continuous at all points when approached from the right. Technically, the formal definition is similar to the definition above for a continuous function but modified as follows:

The function f is said to be right-continuous at the point c if and only if the following holds: For any number ε > 0 however small, there exists some number δ > 0 such that for all x in the domain with c < x < c + δ, the value of f(x) will satisfy

Likewise a left-continuous function is a function which is continuous at all points when approached from the left.

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

Continuous functions between metric spaces

Now consider a function f from one metric space (X, d_X) to another metric space (Y, d_Y). In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined Then f is continuous at the point c in X if for any positive real number ε, there exists a positive real number δ such that all x in X satisfying d_X(x, c) < δ will also satisfy d_Y(f(x), f(c)) < ε.

This can also be formulated in terms of sequences and limits: the function f is continuous at the point c if for every sequence (x_n) in X with limit lim x_n = c, we have lim f(x_n) = f(c). In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Continuous functions transform limits into limits.

This latter condition can be weakened as follows: f is continuous at the point c if and only if for every convergent sequence (x_n) in X with limit c, the sequence (f(x_n)) is a Cauchy sequence, and c is in the domain of f. In Mathematics, a Cauchy sequence, named after Augustin Cauchy, is a Sequence whose elements become arbitrarily close to each other as the sequence Continuous functions transform convergent sequences into Cauchy sequences.

Continuous functions between topological spaces

Main article: Continuous function (topology)

The above definitions of continuous functions can be generalized to functions from one topological space to another in a natural way; a function f : X → Y, where X and Y are topological spaces, is continuous if and only if for every open set V ⊆ Y, the inverse image

is open. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. ↔ In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in

However, this definition is often difficult to use directly. Instead, suppose we have a function f from X to Y, where X,Y are topological spaces. We say f is continuous at x for some if for any neighborhood V of f(x), there is a neighborhood U of x such that . In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. Although this definition appears complex, the intuition is that no matter how "small" V becomes, we can always find a U containing x that will map inside it. If f is continuous at every , then we simply say f is continuous.

In a metric space, it is equivalent to consider the neighbourhood system of open balls centered at x and f(x) instead of all neighborhoods. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Topology and related areas of Mathematics, the neighbourhood system or neighbourhood filter \mathcal{V}(x for a point x is the In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric This leads to the standard ε-δ definition of a continuous function from real analysis, which says roughly that a function is continuous if all points close to x map to points close to f(x). This only really makes sense in a metric space, however, which has a notion of distance.

Note, however, that if the target space is Hausdorff, 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). In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space At an isolated point, every function is continuous.

Continuous functions between partially ordered sets

In order theory, continuity of a function between posets is Scott continuity. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, a function between two Partially ordered sets P and Q is Scott-continuous (named after the mathematician Dana Let X be a complete lattice, then a function f : X → X is continuous if, for each subset Y of X, we have sup f(Y) = f(sup Y). In Mathematics, a complete lattice is a Partially ordered set in which all subsets have both a Supremum (join and an Infimum (meet

Continuous binary relation

A binary relation R on A is continuous if R(a, b) whenever there are sequences (a^k)_i and (b^k)_i in A which converge to a and b respectively for which R(a^k, b^k) for all k. This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations Clearly, if one treats R as a characteristic function in three variables, this definition of continuous is identical to that for continuous functions.

See also

References

1. ^ Heine continuity implies Cauchy continuity without the Axiom of Choice. For the notion of upper or lower semicontinuous Multivalued function see Hemicontinuity In Mathematical analysis, semi-continuity In Mathematics, a piecewise-defined function (also called a piecewise function) is a function whose definition is dependent on the value of the Independent In Mathematical analysis, a function f ( x) is called uniformly continuous if roughly speaking small changes in the input x effect In Mathematics, one may talk about absolute continuity of functions and absolute continuity of measures, and these two notions are closely connected In Mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood In Mathematics, more specifically in Real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions In Mathematics, a function f \mathbb{R} \to \mathbb{R} is symmetrically continuous at a point x if \lim_{h\to 0} f(x+h-f(x-h In Mathematics, a function between two Partially ordered sets P and Q is Scott-continuous (named after the mathematician Dana In Axiomatic set theory, a function f: Ord → Ord is called normal (or a normal function) Iff it is continuous (with In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts In the theory of Stochastic processes there are many senses in which a process may be said to be " continuous " as a function of its "time" parameter Apronus. com.

• Visual Calculus by Lawrence S. Husch, University of Tennessee (2001)