An absolutely continuous function , where , is a function such that for every there is a for which

whenever are finitely many subintervals whose interiors are pairwise disjoint sets and . The notation denotes the class of such functions.

Equivalently, has a derivative almost everywhere, this derivative is Lebesgue integrable, and

The equality must hold for every , and the integral is a Lebesgue integral. This characterization is a form of the fundamental theorems of calculus.

Every Lipschitz function on is absolutely continuous, and every absolutely continuous function has uniform continuity. Neither converse holds. For example, is absolutely continuous but is not a Lipschitz function on . The Cantor function has uniform continuity but is not absolutely continuous.

See also

Absolutely Continuous, Cantor Function, Fundamental Theorems of Calculus, Lebesgue Integral, Lipschitz Function, Uniform Continuity

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References

Hunter, J. K. "Absolutely Continuous Functions." §3.A.2 in Notes on Partial Differential Equations. pp. 78-80. https://www.math.ucdavis.edu/~hunter/pdes/pde_notes.pdf.

Cite this as:

Weisstein, Eric W. "Absolutely Continuous Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbsolutelyContinuousFunction.html

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