In mathematics, the terms continuity, continuous, and continuum are used in a variety of related ways.

Continuity of functions and measures

Continuous function 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 preci ...

* Absolutely continuous function * Absolute continuity of a measure with respect to another measure *

Continuous probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...

: Sometimes this term is used to mean a probability distribution whose

cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...

(c.d.f.) is (simply) continuous. Sometimes it has a less inclusive meaning: a distribution whose c.d.f. is

absolutely continuous In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship betwe ...

with respect to

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

. This less inclusive sense is equivalent to the condition that every set whose Lebesgue measure is 0 has probability 0. *

Geometric continuity In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives (''differentiability class)'' it has over its Domain o ...

Parametric continuity In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; t ...

Continuum

Continuum (set theory) In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by \mathfrak. Georg Cantor proved that the cardinality \mathfrak is larger than the smallest infinity, namely, ...

, the

real line A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced mark ...

or the corresponding cardinal number *

Linear continuum In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line. Formally, a linear continuum is a linearly ordered set ''S'' of more than one element that is densely ordered, i.e., between any t ...

, any ordered set that shares certain properties of the real line *

Continuum (topology) In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the st ...

, a nonempty compact connected

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

(sometimes a

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...

) *

Continuum hypothesis In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...

, a conjecture of Georg Cantor that there is no cardinal number between that of countably infinite sets and the cardinality of the set of all real numbers. The latter cardinality is equal to the cardinality of the set of all subsets of a countably infinite set. *

Cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \bold\mathfrak c (lowercase Fraktur "c") or \ ...

, a cardinal number that represents the size of the set of real numbers

Continuity of functions and measures

Continuum

See also