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# Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. [1]

## Absolute continuity

In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity.

## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

## Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.

## Axiom of countable choice

The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable collection of non-empty sets must have a choice function.

## Édouard Goursat

Édouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his Cours d'analyse mathématique, which appeared in the first decade of the twentieth century.

## Ball (mathematics)

In mathematics, a ball is the space bounded by a sphere.

## Base (topology)

In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B.We are using a convention that the union of empty collection of sets is the empty set.

## Bernard Bolzano

Bernard Bolzano (born Bernardus Placidus Johann Nepomuk Bolzano; 5 October 1781 – 18 December 1848) was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his antimilitarist views.

## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Bounded operator

In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).

## Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

## Camille Jordan

Marie Ennemond Camille Jordan (5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse.

## Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

## Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.

## Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

## Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

## Characterizations of the category of topological spaces

In mathematics, a topological space is usually defined in terms of open sets.

## Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share.

## Classification of discontinuities

Continuous functions are of utmost importance in mathematics, functions and applications.

## Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.

## Closure (topology)

In mathematics, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of S may equivalently be defined as the union of S and its boundary, and also as the intersection of all closed sets containing S. Intuitively, the closure can be thought of as all the points that are either in S or "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.

## Closure operator

In mathematics, a closure operator on a set S is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of S to itself which satisfies the following conditions for all sets X,Y\subseteq S |- | X \subseteq \operatorname(X) | (cl is extensive) |- | X\subseteq Y \Rightarrow \operatorname(X) \subseteq \operatorname(Y) | (cl is increasing) |- | \operatorname(\operatorname(X)).

## Coarse function

In mathematics, coarse functions are functions that may appear to be continuous at a distance, but in reality are not necessarily continuous.

## Codomain

In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall.

## Compact space

In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).

## Comparison of topologies

In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set.

## Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

## Constant function

In mathematics, a constant function is a function whose (output) value is the same for every input value.

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Continuous stochastic process

In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter.

## Cours d'Analyse

Cours d'Analyse de l’École Royale Polytechnique; I.re Partie.

## Curve

In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Descriptive set theory

In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" subsets of the real line and other Polish spaces.

## Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

## Dini continuity

In mathematical analysis, Dini continuity is a refinement of continuity.

## Directed set

In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound.

## Discrete space

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Domain theory

Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains.

## Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.

## Duality (mathematics)

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself.

## Eduard Heine

Heinrich Eduard Heine (16 March 1821, Berlin &ndash; October 1881, Halle) was a German mathematician.

## Equicontinuity

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 a precise sense described herein.

## Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

## Existence theorem

In mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s)..', or more generally 'for all,,...

## Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

## Exponentiation

Exponentiation is a mathematical operation, written as, involving two numbers, the base and the exponent.

## Extreme value theorem

In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval, then f must attain a maximum and a minimum, each at least once.

## F-space

In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d: V × V → R so that.

## Final topology

In general topology and related areas of mathematics, the final topology (or strong, colimit, coinduced, or inductive topology) on a set X, with respect to a family of functions into X, is the finest topology on X which makes those functions continuous.

## First-countable space

In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability".

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

## Function of a real variable

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers, or a subset of that contains an interval of positive length.

## Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

## Functor

In mathematics, a functor is a map between categories.

## Gδ set

In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets.

## Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

## Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods.

## Hölder condition

In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, α, such that for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces.

## Heaviside step function

The Heaviside step function, or the unit step function, usually denoted by or (but sometimes, or), is a discontinuous function named after Oliver Heaviside (1850–1925), whose value is zero for negative argument and one for positive argument.

## Height

Height is the measure of vertical distance, either how "tall" something or someone is, or how "high" the position is.

## Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bi continuous function is a continuous function between topological spaces that has a continuous inverse function.

## Hyperreal number

The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities.

## Identity function

Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.

## Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

## Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.

## Infimum and supremum

In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.

## Infinitesimal

In mathematics, infinitesimals are things so small that there is no way to measure them.

## Initial topology

In general topology and related areas of mathematics, the initial topology (or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on X that makes those functions continuous.

## Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

## Interior (topology)

In mathematics, specifically in topology, the interior of a subset S of points of a topological space X consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions.

## Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval,, as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

## Interval (mathematics)

In mathematics, a (real) 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.

## Inverse function

In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.

## Isolated point

In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S but there exists a neighborhood of x which does not contain any other points of S. This is equivalent to saying that the singleton is an open set in the topological space S (considered as a subspace of X).

## Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

## Kolmogorov space

In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space (named after Andrey Kolmogorov) if for every pair of distinct points of X, at least one of them has a neighborhood not containing the other.

## Kuratowski closure axioms

In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set.

## Limit (category theory)

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.

## Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) "approaches" as the input (or index) "approaches" some value.

## Limit of a function

Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1.

## Limit of a sequence

As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.

## Limit point

In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself.

## Limit superior and limit inferior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence.

## Lindelöf space

In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

## Logarithm

In mathematics, the logarithm is the inverse function to exponentiation.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Metric (mathematics)

In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.

## Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

## Microcontinuity

In non-standard analysis, a discipline within classical mathematics, microcontinuity (or S-continuity) of an internal function f at a point a is defined as follows: Here x runs through the domain of f. In formulas, this can be expressed as follows: For a function f defined on \mathbb, the definition can be expressed in terms of the halo as follows: f is microcontinuous at c\in\mathbb if and only if f(hal(c))\subset hal(f(c)), where the natural extension of f to the hyperreals is still denoted f. Alternatively, the property of microcontinuity at c can be expressed by stating that the composition \text\circ f is constant of the halo of c, where "st" is the standard part function.

## Neighbourhood (mathematics)

In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.

## Neighbourhood system

In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x is the collection of all neighbourhoods for the point x.

## Net (mathematics)

In mathematics, more specifically in general topology and related branches, a net or Moore&ndash;Smith sequence is a generalization of the notion of a sequence.

## Non-standard analysis

The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.

## Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

## Normal function

In axiomatic set theory, a function f: Ord → Ord is called normal (or a normal function) iff it is continuous (with respect to the order topology) and strictly monotonically increasing.

## Normed vector space

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined.

## Nowhere continuous function

In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain.

## Open and closed maps

In topology, an open map is a function between two topological spaces which maps open sets to open sets.

## Open set

In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

## Order theory

Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations.

## Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.

## Oscillation (mathematics)

In mathematics, the oscillation of a function or a sequence is a number that quantifies how much a sequence or function varies between its extreme values as it approaches infinity or a point.

## Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

## Pathological (mathematics)

In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved.

## Peter Gustav Lejeune Dirichlet

Johann Peter Gustav Lejeune Dirichlet (13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a function.

## Picard–Lindelöf theorem

In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.

## Piecewise

In mathematics, a piecewise-defined function (also called a piecewise function or a hybrid function) is a function defined by multiple sub-functions, each sub-function applying to a certain interval of the main function's domain, a sub-domain.

## Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

## Preference (economics)

In economics and other social sciences, preference is the ordering of alternatives based on their relative utility, a process which results in an optimal "choice" (whether real or theoretical).

## Quantale

In mathematics, quantales are certain partially ordered algebraic structures that generalize locales (point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann algebras).

## Quotient space (topology)

In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Removable singularity

In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.

## Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.

## Scott continuity

In mathematics, given two partially ordered sets P and Q, a function f \colon P \rightarrow Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema, i.e. if for every directed subset D of P with supremum in P its image has a supremum in Q, and that supremum is the image of the supremum of D: that is, \sqcup f.

## Semi-continuity

In mathematical analysis, semi-continuity (or semicontinuity) is a property of extended real-valued functions that is weaker than continuity.

## Separable space

In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence \_^ of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Sequential space

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

## Sign (mathematics)

In mathematics, the concept of sign originates from the property of every non-zero real number of being positive or negative.

## Sign function

In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number.

## Sinc function

In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by, has two slightly different definitions.

## Sine

In mathematics, the sine is a trigonometric function of an angle.

## Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## Square root

In mathematics, a square root of a number a is a number y such that; in other words, a number y whose square (the result of multiplying the number by itself, or) is a. For example, 4 and −4 are square roots of 16 because.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a topology induced from that of X called the subspace topology (or the relative topology, or the induced topology, or the trace topology).

## Surjective function

In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).

## Symmetrically continuous function

In mathematics, a function f: \mathbb \to \mathbb is symmetrically continuous at a point x The usual definition of continuity implies symmetric continuity, but the converse is not true.

## Thomae's function

Thomae's function, named after Carl Johannes Thomae, has many names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function, the Riemann function, or the Stars over Babylon (John Horton Conway's name).

## Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

## Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

## Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

## Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

## Trivial topology

In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space.

## Uniform continuity

In mathematics, a function f is uniformly continuous if, roughly speaking, it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, the maximum distance between f(x) and f(y) cannot depend on x and y themselves.

## Uniform convergence

In the mathematical field of analysis, uniform convergence is a type of convergence of functions stronger than pointwise convergence.

## Uniform space

In the mathematical field of topology, a uniform space is a set with a uniform structure.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## Weierstrass function

In mathematics, the Weierstrass function is an example of a pathological real-valued function on the real line.

## (ε, δ)-definition of limit

In calculus, the (ε, δ)-definition of limit ("epsilon–delta definition of limit") is a formalization of the notion of limit.

## 0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

## References

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