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Absolutely Continuous

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Absolutely continuous real-numbered functions are those functions for which the Fundamental Theorem of Calculus (FTC) holds [1]. In other words, absolute continuity identifies which functions can be antiderivatives: a function on a closed, bounded interval is absolutely continuous on that interval if it is also an antiderivative over that same interval [2].

These functions have the “smoothest” type of continuity, followed by uniform continuity and then plain old continuity. All absolutely continuous functions are continuous, but the converse is not true.

Absolutely continuous functions and random variables are related to each other in the following way: A real-valued random variable X is absolutely continuous if its distribution function FX is absolutely continuous [3]


Formal Definition

The formal definition is frequently used in real analysis, particularly for proving the Fundamental Theorem of Calculus for the Lebesgue Integral [4].

An absolutely continuous function, defined on a closed interval, has the following property. The property is based on a positive number ε and its counterpart, another positive number δ.

  1. Take the interval for which we want to define absolute continuity, then break it into a set of finite, nonoverlapping intervals. The lengths of these intervals have a sum less than δ,
  2. Next, consider the absolute values of the differences in the function values at the ends of the intervals; The sum over these intervals is less than ε [5].

In summation notation, we can state the above as: when a finite sequence of non-overlapping intervals satisfies:


then


Where (xk, yk) are the non-overlapping subintervals.

Examples and Properties of Absolutely Continuous Functions

Every convex function and every continuously differentiable function is absolutely continuous [3].


Given a real-valued absolutely continuous function, the following properties hold [6]:

  • cf, where c ∈ ℝ
  • f + g
  • fg
  • 1/f, if f(x) ≠ 0 for every x ∈ [a, b]
  • |f|.

A few specific examples: The Lipschitz function is absolutely continuous; The Cantor function, is not (although it is continuous everywhere) [7]. The function tan(x) is neither uniformly continuous nor absolutely continuous on the interval [0, π/2].

References

[1] Heil, C. Real Analysis Lecture notes. 3.5. Abs. cont. and singular functions. Retrieved May 4, 2021 from: https://people.math.gatech.edu/~heil/handouts/ac.pdf
[2] 7.4 Abs. Cont. & Singular Functions. Retrieved May 4, 2021 from: https://www.math.lsu.edu/~rich/Absolute_Continuity
[3] Hill, T. & Berger, A. (2015). An Introduction to Benford’s Law. Princeton University Press.
[4] Pouso, R. (2012). A simple proof of the Fundamental Theorem of Calculus for the Lebesgue integral. Retrieved May 5, 2021 from: https://arxiv.org/pdf/1203.1462.pdf
[5] McGraw-Hill Dictionary of Mathematics, 2/E. (2002). McGraw-Hill.
[6] World Scientific. (2014). Problems and proofs in real analysis. pp. 314-352. Retrieved May 5, 2021 from: https://www.worldscientific.com/doi/10.1142/9789814578516_0013
[7] Real Analysis January 9, 2016 Chapter 6. Differentiation and Integration. Retrieved May 4, 2021 from: https://faculty.etsu.edu/gardnerr/5210/Beamer-Proofs/Proofs-6-5.pdf

CITE THIS AS:
Stephanie Glen. "Absolutely Continuous" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/absolutely-continuous/
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