absolutely continuous function

is the precise condition one needs to impose in order for the fundamental theorem of calculusMathworldPlanetmathPlanetmath to hold for the Lebesgue integralMathworldPlanetmath.

Definition Suppose [a,b] be a closed bounded interval of . Then a function f:[a,b] is absolutely continuousMathworldPlanetmath on [a,b], if for any ε>0, there is a δ>0 such that the following condition holds:

Theorem 1 ().

Let f:[a,b]C be a function. Then f is absolutely continuous if and only if there is a function gL1(a,b) (i.e. a g:(a,b)C with ab|g|<), such that


for all x[a,b]. What is more, if f and g are as above, then f is differentiableMathworldPlanetmathPlanetmath almost everywhere and f=g almost everywhere. (Above, both integrals are Lebesgue integrals.)

See [2, 3] for proof.

See also [1], and [4] for a discussion about different proofs.


  • 1 Wikipedia, entry on http://en.wikipedia.org/wiki/Absolute_continuityAbsolute continuity.
  • 2 F. Jones, Lebesgue Integration on Euclidean Spaces, Jones and Barlett Publishers, 1993.
  • 3 C.D. Aliprantis, O. Burkinshaw, Principles of Real Analysis, 2nd ed., Academic Press, 1990.
  • 4 D. B’arcenas, The Fundamental Theorem of Calculus for Lebesgue Integral, Divulgaciones Matemáticas, Vol. 8, No. 1, 2000, pp. 75-85.
Title absolutely continuous function
Canonical name AbsolutelyContinuousFunction
Date of creation 2013-03-22 15:18:47
Last modified on 2013-03-22 15:18:47
Owner matte (1858)
Last modified by matte (1858)
Numerical id 13
Author matte (1858)
Entry type Definition
Classification msc 26B30
Classification msc 26A46
Related topic SingularFunction
Related topic AbsolutelyContinuous
Defines fundamental theorem of calculus for the Lebesgue integral