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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