Approximately continuous function: Difference between revisions

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{{Short description|Mathematical concept in measure theory}}

{{Multiple issues|

{{Page numbers needed|date=January 2025}}

{{technical|date=May 2025}}

In [[mathematics]], particularly in [[mathematical analysis]] and [[measure theory]], an '''approximately right function''' is a concept that generalizes the notion of [[continuous function]]s by replacing the [[limit of a function|ordinary limit]] with an [[approximate limit]].<ref>{{cite web|url=https://encyclopediaofmath.org/wiki/Approximate_continuity|title=Approximate continuity|website=Encyclopedia of Mathematics|access-date=January 7, 2025}}</ref> This generalization provides insights into [[measurable function]]s with applications in real analysis and geometric measure theory.<ref>{{cite book |last1=Evans |first1=L.C. |last2=Gariepy |first2=R.F. |title=Measure theory and fine properties of functions |publisher=CRC Press |series=Studies in Advanced Mathematics |___location=Boca Raton, FL |year=1992 |isbn= |pages=}}</ref>

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In [[mathematics]], particularly in [[mathematical analysis]] and [[measure theory]], an '''approximately rightcontinuous function''' is a concept that generalizes the notion of [[continuous function]]s by replacing the [[limit of a function|ordinary limit]] with an [[approximate limit]].<ref>{{cite web|url=https://encyclopediaofmath.org/wiki/Approximate_continuity|title=Approximate continuity|website=Encyclopedia of Mathematics|access-date=January 7, 2025}}</ref> This generalization provides insights into [[measurable function]]s with applications in real analysis and geometric measure theory.<ref>{{cite book |last1=Evans |first1=L.C. |last2=Gariepy |first2=R.F. |title=Measure theory and fine properties of functions |publisher=CRC Press |series=Studies in Advanced Mathematics |___location=Boca Raton, FL |year=1992 |isbn= |pages=}}</ref>

== Definition ==

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<blockquote>

'''Stepanov-Denjoy theorem:''' A function is [[measurable function|measurable]] [[if and only if]] it is approximately continuous [[almost everywhere]].

<ref>{{cite journal| issn = 0528-2195| volume = 103| issue = 1| pages = 95–96| last = Lukeš| first = Jaroslav| title = A topological proof of Denjoy-Stepanoff theorem| journal = Časopis pro pěstování matematiky| access-date = 2025-01-20| date = 1978| doi = 10.21136/CPM.1978.117963| url = https://dml.cz/handle/10338.dmlcz/117963| doi-access = free}}</ref>

</blockquote>

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* [[Approximate limit]]

* [[Density point]]

* [[Density topology]] (which serves to describe approximately continuous functions in a different way, as continuous functions for a different topology)

* [[Lebesgue point]]

* [[Lusin's theorem]]