{{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 continuous function''' is a concept that generalizes the notion of continuous functions by replacing the 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 functions 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-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> '''Stepanov-Denjoy theorem:''' A function is 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| hdl = 10338.dmlcz/117963| hdl-access = free}}</ref> </blockquote>
Approximately continuous functions are intimately connected to Lebesgue points. 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 * Measurable function
== References == {{reflist}}
Category:Theory of continuous functions Category:Calculus Category:Real analysis Category:Mathematical analysis Category:Measure theory Category:Types of functions