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

## Definition

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

## Equivalent Conditions

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.

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