Approximately continuous function: Difference between revisions

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== 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:

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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 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| url = https://dml.cz/handle/10338.dmlcz/117963}}</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