Lebesgue’s Theorem (see below) has been called one of the most striking theorems in real analysis. Indeed it is a very surprising result.
Lebesgue’s Theorem (Monotone functions)
If the function is monotone on the open interval , then it is differentiable almost everywhere on .
Absolutely Continuous Functions
A real-valued function on a closed, bounded interval is said to be absolutely continuous on provided for each , there is a such that for every finite disjoint collection of open intervals in , if then
The following conditions on a real-valued function on a compact interval are equivalent:
(i) is absolutely continuous;
(ii) has a derivative almost everywhere, the derivative is Lebesgue integrable, and for all on ;
(iii) there exists a Lebesgue integrable function on such that for all on .
Equivalence between (i) and (iii) is known as the Fundamental Theorem of Lebesgue integral calculus.