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

Continuity of functions and measures

* Continuous function * 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 the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

: Sometimes this term is used to mean a probability distribution whose cumulative distribution function (c.d.f.) is (simply) continuous. Sometimes it has a less inclusive meaning: a distribution whose c.d.f. is

absolutely continuous In calculus, 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 between the two central ope ...

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

Parametric 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 it has over some domain, called ''differentiability cl ...

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 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 (sometimes a

Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...

) *

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

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

Continuity of functions and measures

Continuum

See also