Approximately continuous function

Let $E\; \backslash subseteq\; \backslash mathbb\{R\}^n$ be a Lebesgue measurable set, $f\backslash colon\; E\; \backslash to\; \backslash mathbb\{R\}^k$ be a measurable function, and $x\_0\; \backslash in\; E$ be a point where the Lebesgue density of $E$ is 1. The function $f$ is said to be approximately continuous at $x\_0$ if and only if the approximate limit of $f$ at $x\_0$ exists and equals $f(x\_0)$.{{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=}}

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.{{cite book |last=Saks |first=S. |title=Theory of the integral |publisher=Hafner |year=1952 |isbn= |pages=}} 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:

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

{{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}}

Approximately continuous functions are intimately connected to Lebesgue points. For a function $f\; \backslash in\; L^1(E)$, a point $x\_0$ is a Lebesgue point if it is a point of Lebesgue density 1 for $E$ and satisfies

:$\backslash lim\_\{r\backslash downarrow\; 0\}\; \backslash frac\{1\}\{\backslash lambda\; (B\_r\; (x\_0))\}\; \backslash int\_\{E\backslash cap\; B\_r\; (x\_0)\}\; |f(x)-f(x\_0)|\backslash ,\; dx\; =\; 0$

where $\backslash lambda$ denotes the Lebesgue measure and $B\_r(x\_0)$ represents the ball of radius $r$ centered at $x\_0$. Every Lebesgue point of a function is necessarily a point of approximate continuity.{{cite book |last=Thomson |first=B.S. |title=Real functions |publisher=Springer |year=1985 |isbn= |pages=}} The converse relationship holds under additional constraints: when $f$ is essentially bounded, its points of approximate continuity coincide with its Lebesgue points.{{cite book |last=Munroe |first=M.E. |title=Introduction to measure and integration |publisher=Addison-Wesley |year=1953 |isbn= |pages=}}

• 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

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Category:Theory of continuous functions

Category:Calculus

Category:Real analysis

Category:Mathematical analysis

Category:Measure theory

Category:Types of functions