Approximately continuous function
 | This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages)
|
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. This generalization provides insights into measurable functions with applications in real analysis and geometric measure theory.
Definition
Let be a Lebesgue measurable set, be a measurable function, and be a point where the Lebesgue density of is 1. The function is said to be approximately continuous at if and only if the approximate limit of at exists and equals .
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. The concept of approximate continuity can be extended beyond measurable functions to arbitrary functions between metric spaces. The Stepanov-Denjoy theorem states:
Stepanov-Denjoy theorem: A function is measurable if and only if it is approximately continuous almost everywhere.
Approximately continuous functions are intimately connected to Lebesgue points. For a function , a point is a Lebesgue point if it is a point of Lebesgue density 1 for and satisfies where denotes the Lebesgue measure and represents the ball of radius centered at . Every Lebesgue point of a function is necessarily a point of approximate continuity. The converse relationship holds under additional constraints: when is essentially bounded, its points of approximate continuity coincide with its Lebesgue points.
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
- ↑ "Approximate continuity". Encyclopedia of Mathematics. Retrieved January 7, 2025.
- ↑ Evans, L.C.; Gariepy, R.F. (1992). Measure theory and fine properties of functions. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press.
- ↑ Federer, H. (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. New York: Springer-Verlag.
- ↑ Saks, S. (1952). Theory of the integral. Hafner.
- ↑ Lukeš, Jaroslav (1978). "A topological proof of Denjoy-Stepanoff theorem". Časopis pro pěstování matematiky. 103 (1): 95–96. doi:10.21136/CPM.1978.117963. hdl:10338.dmlcz/117963. ISSN 0528-2195. Retrieved 2025-01-20.
- ↑ Thomson, B.S. (1985). Real functions. Springer.
- ↑ Munroe, M.E. (1953). Introduction to measure and integration. Addison-Wesley.