Approximately continuous function: Difference between revisions

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

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In [[mathematics]], particularly in [[mathematical analysis]] and [[measure theory]], an '''approximately continuous 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 ==

Let <math>E \subseteq \mathbb{R}^n</math> be a [[Lebesgue measurable set]], <math>f\colon E \to \mathbb{R}^k</math> be a [[measurable function]], and <math>x_0 \in E</math> be a point where the [[Lebesgue density]] of <math>E</math> is 1. The function <math>f</math> is said to be ''approximately continuous'' at <math>x_0</math> if and only if the [[approximate limit]] of <math>f</math> at <math>x_0</math> exists and equals <math>f(x_0)</math>.<ref>{{cite book |last=Federer |first=H. |title=Geometric measure theory |publisher=[[Springer Science+Business Media|Springer-Verlag]] |series=Die Grundlehren der mathematischen Wissenschaften |volume=153 |___location=New York |year=1969 |isbn= |pages=}}</ref>

== Properties ==

A fundamental result in the theory of approximately continuous functions is derived from [[Lusin's theorem]], which states that every measurable function is approximately continuous at almost every point of its ___domain.<ref>{{cite book |last=Saks |first=S. |title=Theory of the integral |publisher=Hafner |year=1952 |isbn= |pages=}}</ref> The concept of approximate continuity can be extended beyond measurable functions to arbitrary functions between metric spaces. The '''Stepanov-Denjoy theorem''' provides a remarkable characterization:

<blockquote>

<blockquote>'''Stepanov-Denjoy theorem:''' A function is [[measurable function|measurable]] [[if and only if]] it is approximately continuous [[almost everywhere]].<ref>{{cite book |last=Bruckner |first=A.M. |title=Differentiation of real functions |publisher=Springer |year=1978 |isbn= |pages=}}</ref></blockquote>

<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>

Approximately continuous functions are intimately connected to [[Lebesgue point]]s. For a function <math>f \in L^1(E)</math>, a point <math>x_0</math> is a Lebesgue point if it is a point of Lebesgue density 1 for <math>E</math> and satisfies

:<math>\lim_{r\downarrow 0} \frac{1}{\lambda (B_r (x_0))} \int_{E\cap B_r (x_0)} |f(x)-f(x_0)|\, dx = 0</math>

where <math>\lambda</math> denotes the [[Lebesgue measure]] and <math>B_r(x_0)</math> represents the ball of radius <math>r</math> centered at <math>x_0</math>. Every Lebesgue point of a function is necessarily a point of approximate continuity.<ref>{{cite book |last=Thomson |first=B.S. |title=Real functions |publisher=Springer |year=1985 |isbn= |pages=}}</ref> The converse relationship holds under additional constraints: when <math>f</math> is [[essentially bounded]], its points of approximate continuity coincide with its Lebesgue points.<ref>{{cite book |last=Munroe |first=M.E. |title=Introduction to measure and integration |publisher=[[Addison-Wesley]] |year=1953 |isbn= |pages=}}</ref>

== See also ==

* [[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]]