Continuous function

continuouscontinuitycontinuous mapcontinuous functionscontinuous mappingcontinuouslycontinuous mapsdiscontinuous functiondiscontinuoussequentially continuous
In mathematics, a continuous function is a function that does not have any jumps or other discontinuities.wikipedia
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Topology

topologicaltopologicallytopologist
Continuity of functions is one of the core concepts of topology, which is treated in full generality below.
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.

Homeomorphism

homeomorphichomeomorphismstopologically equivalent
A continuous function with a continuous inverse function is called a homeomorphism.
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function.

Karl Weierstrass

WeierstrassKarl WeierstraßKarl Theodor Wilhelm Weierstrass
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.
Weierstrass formalized the definition of the continuity of a function, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals.

Curve

closed curvespace curvesmooth curve
A real function, that is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line.
This definition of a curve has been formalized in modern mathematics as: A curve is the image of a continuous function from an interval to a topological space.

Function (mathematics)

functionfunctionsmathematical function
A real function, that is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. In mathematics, a continuous function is a function that does not have any jumps or other discontinuities.
Whichever definition of map is used, related terms like domain, codomain, injective, continuous have the same meaning as for a function.

Limit (mathematics)

limitlimitsconverge
A rigorous definition of continuity of real functions is usually given in a first course in calculus in terms of the idea of a limit.
Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

Graph of a function

graphgraphsgraphing
A real function, that is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line.
are real numbers, these pairs are Cartesian coordinates of points in the Euclidean plane and thus form a subset of this plane, which is a curve in the case of a continuous function.

Limit of a function

limitlimitsconvergence
The function f is continuous at some point c 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).
In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function.

Lipschitz continuity

Lipschitz functionLipschitzLipschitz continuous
For example, the Lipschitz and Hölder continuous functions of exponent α below are defined by the set of control functions
Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (or modulus of uniform continuity).

Function of a real variable

real functionreal variablefunctions of a real variable
A real function, that is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line.
Conversely, it is sometimes possible to enlarge naturally the domain of a given function, for example by continuity or by analytic continuation.

Real number

realrealsreal-valued
A real function, that is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states:

Inverse function

inverseinvertibleinvertible function
A continuous function with a continuous inverse function is called a homeomorphism.
A continuous function f is invertible on its range (image) if and only if it is either strictly increasing or decreasing (with no local maxima or minima).

Classification of discontinuities

discontinuitiesdiscontinuousdiscontinuity
In mathematics, a continuous function is a function that does not have any jumps or other discontinuities.
Continuous functions are of utmost importance in mathematics, functions and applications.

Heaviside step function

Heaviside functionunit step functionHeaviside unit step function
An example of a discontinuous function is the Heaviside step function H, defined by
), is a discontinuous function, named after Oliver Heaviside (1850–1925), whose value is zero for negative arguments and one for positive arguments.

Pathological (mathematics)

well-behavedpathologicalbadly behaved
Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined pathological, for example, Thomae's function,
A classic example of a pathological structure is the Weierstrass function, which is continuous everywhere but differentiable nowhere.

Mathematics

mathematicalmathmathematician
In mathematics, a continuous function is a function that does not have any jumps or other discontinuities.
These, in turn, are contained within the real numbers, which are used to represent continuous quantities.

Exponentiation

exponentpowerpowers
In this case, the previous two examples are not continuous, but every polynomial function is continuous, as are the sine, cosine, and exponential functions.
Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by continuity with the rule

Polynomial

polynomial functionpolynomialsmultivariate polynomial
In this case, the previous two examples are not continuous, but every polynomial function is continuous, as are the sine, cosine, and exponential functions.
Every polynomial function is continuous, smooth, and entire.

Intermediate value theorem

Bolzano's theoremBolzano–Cauchy theoremintermediate value property
The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states:
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any value between f(a) and f(b) at some point within the interval.

Infinitesimal

infinitesimalsinfinitely closeinfinitesimally
Cauchy defined continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34).
Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function.

Metric space

metricmetric spacesmetric geometry
In addition, this article discusses the definition for the more general case of functions between two metric spaces.

Extreme value theorem

boundedboundedness theoremboundedness, ultimate boundedness
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.e. there exists c ∈ [a,b] with f(c) ≥ f(x) for all x ∈ [a,b].
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval [a,b], then f must attain a maximum and a minimum, each at least once.

Nowhere continuous function

Dirichlet functionnowhere continuousDirichlet's function
In a similar vein, Dirichlet's function, the indicator function for the set of rational numbers,
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.

Smoothness

smoothsmooth functionsmooth map
See differentiability class.
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

Weierstrass function

functionNowhere differentiable functioncontinuous functions with no derivatives
Weierstrass's function is also everywhere continuous but nowhere differentiable.
The function has the property of being continuous everywhere but differentiable nowhere.