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Approximately continuous function

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

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 provides a remarkable characterization:

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 f ∈ L 1 ( E ) {\displaystyle f\in L^{1}(E)} , a point x 0 {\displaystyle x_{0}} is a Lebesgue point if it is a point of Lebesgue density 1 for E {\displaystyle E} and satisfies

lim r ↓ 0 1 λ ( B r ( x 0 ) ) ∫ E ∩ B r ( x 0 ) | f ( x ) − f ( x 0 ) | d x = 0 {\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 λ {\displaystyle \lambda } denotes the Lebesgue measure and B r ( x 0 ) {\displaystyle B_{r}(x_{0})} represents the ball of radius r {\displaystyle r} centered at x 0 {\displaystyle x_{0}} . Every Lebesgue point of a function is necessarily a point of approximate continuity. The converse relationship holds under additional constraints: when f {\displaystyle f} is essentially bounded, its points of approximate continuity coincide with its Lebesgue points.

See also

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. Bruckner, A.M. (1978). Differentiation of real functions. Springer.
  6. Thomson, B.S. (1985). Real functions. Springer.
  7. Munroe, M.E. (1953). Introduction to measure and integration. Addison-Wesley.
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