Editing Continuous function (section)

==Related notions==

If <math>f : S \to Y</math> is a continuous function from some subset <math>S</math> of a topological space <math>X</math> then a {{em|{{visible anchor|continuous extension|Continuous extension}}}} of <math>f</math> to <math>X</math> is any continuous function <math>F : X \to Y</math> such that <math>F(s) = f(s)</math> for every <math>s \in S,</math> which is a condition that often written as <math>f = F\big\vert_S.</math> In words, it is any continuous function <math>F : X \to Y</math> that [[Restriction of a function|restricts]] to <math>f</math> on <math>S.</math> This notion is used, for example, in the [[Tietze extension theorem]] and the [[Hahn–Banach theorem]]. If <math>f : S \to Y</math> is not continuous, then it could not possibly have a continuous extension. If <math>Y</math> is a [[Hausdorff space]] and <math>S</math> is a [[Dense set|dense subset]] of <math>X</math> then a continuous extension of <math>f : S \to Y</math> to <math>X,</math> if one exists, will be unique. The [[Blumberg theorem]] states that if <math>f : \R \to \R</math> is an arbitrary function then there exists a dense subset <math>D</math> of <math>\R</math> such that the restriction <math>f\big\vert_D : D \to \R</math> is continuous; in other words, every function <math>\R \to \R</math> can be restricted to some dense subset on which it is continuous. 

Various other mathematical domains use the concept of continuity in different but related meanings. For example, in [[order theory]], an order-preserving function <math>f : X \to Y</math> between particular types of [[partially ordered set]]s <math>X</math> and <math>Y</math> is continuous if for each [[Directed set|directed subset]] <math>A</math> of <math>X,</math> we have <math>\sup f(A) = f(\sup A).</math> Here <math>\,\sup\,</math> is the [[supremum]] with respect to the orderings in <math>X</math> and <math>Y,</math> respectively. This notion of continuity is the same as topological continuity when the partially ordered sets are given the [[Scott topology]].<ref>{{cite book |last=Goubault-Larrecq |first=Jean |title=Non-Hausdorff Topology and Domain Theory: Selected Topics in Point-Set Topology |publisher=[[Cambridge University Press]]|year=2013 |isbn=978-1107034136}}</ref><ref>{{cite book |last1=Gierz |first1=G. |last2=Hofmann |first2=K. H. |last3=Keimel |first3=K. |last4=Lawson |first4=J. D. |last5=Mislove |first5=M. W. |last6=Scott |first6=D. S. |title=Continuous Lattices and Domains |volume=93 |series=Encyclopedia of Mathematics and its Applications |publisher=Cambridge University Press |year=2003 |isbn=0521803381 |url-access=registration |url=https://archive.org/details/continuouslattic0000unse}}</ref>

In [[category theory]], a [[functor]]
<math display="block">F : \mathcal C \to \mathcal D</math>
between two [[Category (mathematics)|categories]] is called {{em|[[Continuous functor|continuous]]}} if it commutes with small [[Limit (category theory)|limits]]. That is to say,
<math display="block">\varprojlim_{i \in I} F(C_i) \cong F \left(\varprojlim_{i \in I} C_i \right)</math>
for any small (that is, indexed by a set <math>I,</math> as opposed to a [[class (mathematics)|class]]) [[Diagram (category theory)|diagram]] of [[Object (category theory)|objects]] in <math>\mathcal C</math>.

A {{em|[[continuity space]]}} is a generalization of metric spaces and posets,<ref>{{cite journal | title = Quantales and continuity spaces | citeseerx=10.1.1.48.851 | first = R. C. | last =Flagg | journal = Algebra Universalis | year = 1997 | volume=37 | issue=3 | pages=257–276 | doi=10.1007/s000120050018 | s2cid=17603865 }}</ref><ref>{{cite journal | title = All topologies come from generalized metrics | first = R. | last = Kopperman | journal =  American Mathematical Monthly | year = 1988 |volume=95 |issue=2 |pages=89–97 |doi=10.2307/2323060 | jstor = 2323060 }}</ref> which uses the concept of [[quantale]]s, and that can be used to unify the notions of metric spaces and [[Domain theory|domain]]s.<ref>{{cite journal | title = Continuity spaces: Reconciling domains and metric spaces | first1 = B. | last1 = Flagg | first2 = R. | last2 = Kopperman | journal = Theoretical Computer Science |volume=177 |issue=1 |pages=111–138 |doi=10.1016/S0304-3975(97)00236-3 | year = 1997 | doi-access = free }}</ref>

In [[measure theory]], a function <math>f : E \to \mathbb{R}^k</math> defined on a [[Lebesgue measurable set]] <math>E \subseteq \mathbb{R}^n</math> is called [[approximately continuous]] at a point <math>x_0 \in E</math> if the [[approximate limit]] of <math>f</math> at <math>x_0</math> exists and equals <math>f(x_0)</math>. This generalizes the notion of continuity by replacing the ordinary limit with the [[approximate limit]]. A fundamental result known as the [[Stepanov-Denjoy theorem]] states that a function is [[measurable function|measurable]] if and only if it is approximately continuous [[almost everywhere]].<ref>{{cite book |last=Federer |first=H. |title=Geometric measure theory |publisher=Springer-Verlag |series=Die Grundlehren der mathematischen Wissenschaften |volume=153 |location=New York |year=1969 |isbn= |pages=}}</ref>