absolutely continuous function
is the precise condition one needs to impose in order for the fundamental theorem of calculus^{} to hold for the Lebesgue integral^{}.
Definition Suppose $[a,b]$ be a closed bounded interval of $\mathbb{R}$. Then a function $f:[a,b]\to \u2102$ is absolutely continuous^{} on $[a,b]$, if for any $\epsilon >0$, there is a $\delta >0$ such that the following condition holds:

($\ast $)
If $({a}_{1},{b}_{1}),\mathrm{\dots},({a}_{n},{b}_{n})$ is a finite collection^{} of disjoint open intervals in $[a,b]$ such that
$$ then
$$
Theorem 1 ().
Let $f\mathrm{:}\mathrm{[}a\mathrm{,}b\mathrm{]}\mathrm{\to}\mathrm{C}$ be a function. Then $f$ is absolutely continuous if and only if there is a function $g\mathrm{\in}{L}^{\mathrm{1}}\mathit{}\mathrm{(}a\mathrm{,}b\mathrm{)}$ (i.e. a $g\mathrm{:}\mathrm{(}a\mathrm{,}b\mathrm{)}\mathrm{\to}\mathrm{C}$ with $$), such that
$$f(x)=f(a)+{\int}_{a}^{x}g(t)\mathit{d}t$$ 
for all $x\mathrm{\in}\mathrm{[}a\mathrm{,}b\mathrm{]}$. What is more, if $f$ and $g$ are as above, then $f$ is differentiable^{} almost everywhere and ${f}^{\mathrm{\prime}}\mathrm{=}g$ almost everywhere. (Above, both integrals are Lebesgue integrals.)
References
 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. 7585.
Title  absolutely continuous function 

Canonical name  AbsolutelyContinuousFunction 
Date of creation  20130322 15:18:47 
Last modified on  20130322 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 