The most Striking Theorem in Real Analysis

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 $f$ is monotone on the open interval $(a,b)$ , then it is differentiable almost everywhere on $(a,b)$ .

Absolutely Continuous Functions

Definition

A real-valued function $f$ on a closed, bounded interval $[a,b]$ is said to be absolutely continuous on $[a,b]$ provided for each $\epsilon>0$ , there is a $\delta>0$ such that for every finite disjoint collection $\{(a_k,b_k)\}_{k=1}^n$ of open intervals in $(a,b)$ , if $\displaystyle \sum_{k=1}^n(b_k-a_k)<\delta,$ then $\displaystyle \sum_{k=1}^n|f(b_k)-f(a_k)|<\epsilon.$

Equivalent Conditions

The following conditions on a real-valued function $f$ on a compact interval $[a,b]$ are equivalent:
(i) $f$ is absolutely continuous;

(ii) $f$ has a derivative $f'$ almost everywhere, the derivative is Lebesgue integrable, and $\displaystyle f(x)=f(a)+\int_a^x f'(t)\,dt$ for all $x$ on $[a,b]$ ;

(iii) there exists a Lebesgue integrable function $g$ on $[a,b]$ such that $\displaystyle f(x)=f(a)+\int_a^x g(t)\,dt$ for all $x$ on $[a,b]$ .

Equivalence between (i) and (iii) is known as the Fundamental Theorem of Lebesgue integral calculus.