Approximately continuous function

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.^[1] This generalization provides insights into measurable functions with applications in real analysis and geometric measure theory.^[2]

Definition

Let ${\displaystyle E\subseteq \mathbb {R} ^{n}}$ be a Lebesgue measurable set, ${\displaystyle f\colon E\to \mathbb {R} ^{k}}$ be a measurable function, and ${\displaystyle x_{0}\in E}$ be a point where the Lebesgue density of ${\displaystyle E}$ is 1. The function ${\displaystyle f}$ is said to be approximately continuous at ${\displaystyle x_{0}}$ if and only if the approximate limit of ${\displaystyle f}$ at ${\displaystyle x_{0}}$ exists and equals ${\displaystyle f(x_{0})}$.^[3]

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.^[4] 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. ^[5]

Approximately continuous functions are intimately connected to Lebesgue points. For a function ${\displaystyle f\in L^{1}(E)}$, a point ${\displaystyle x_{0}}$ is a Lebesgue point if it is a point of Lebesgue density 1 for ${\displaystyle E}$ and satisfies

${\displaystyle \lim _{r\downarrow 0}{\frac {1}{\lambda (B_{r}(x_{0}))}}\int _{E\cap B_{r}(x_{0})}|f(x)-f(x_{0})|\,dx=0}$

where ${\displaystyle \lambda }$ denotes the Lebesgue measure and ${\displaystyle B_{r}(x_{0})}$ represents the ball of radius ${\displaystyle r}$ centered at ${\displaystyle x_{0}}$. Every Lebesgue point of a function is necessarily a point of approximate continuity.^[6] The converse relationship holds under additional constraints: when ${\displaystyle f}$ is essentially bounded, its points of approximate continuity coincide with its Lebesgue points.^[7]

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

1. "Approximate continuity". Encyclopedia of Mathematics. Retrieved January 7, 2025.
2. Evans, L.C.; Gariepy, R.F. (1992). Measure theory and fine properties of functions. Studies in Advanced Mathematics. Boca Raton, FL: CRC Press.
3. Federer, H. (1969). Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften. Vol. 153. New York: Springer-Verlag.
4. Saks, S. (1952). Theory of the integral. Hafner.
5. Lukeš, Jaroslav (1978). "A topological proof of Denjoy-Stepanoff theorem". Časopis pro pěstování matematiky. 103 (1): 95–96. doi:10.21136/CPM.1978.117963. ISSN 0528-2195. Retrieved 2025-01-20.
6. Thomson, B.S. (1985). Real functions. Springer.
7. Munroe, M.E. (1953). Introduction to measure and integration. Addison-Wesley.