Lebesgue's theorem on differentiability of monotone functions

Remarks

The standard proof uses the Dini derivatives of $f$ and the Vitali covering lemma. For nondecreasing $f$, one shows that each set $\set{x : D^+ f(x) > q > p > D_- f(x)}$ with rationals $p < q$ is null. One also shows that the set where $D^+ f(x) = \infty$ is null. Together with the same argument for the other pairs of Dini derivatives, this gives a finite derivative almost everywhere.

For nondecreasing $f$, the derivative $f'$ is moreover Lebesgue integrable, and $\int_a^b f'(x) \, dx \le f(b) - f(a)$ for the Lebesgue integral. The inequality can be strict: for the Cantor function, the left side is $0$ and the right side is $1$. Monotonicity cannot be replaced by continuity: the Weierstrass function is continuous and differentiable at no point. The theorem extends to functions of bounded variation, which are differences of monotone functions by the Jordan decomposition.

Going deeper

For a student

The theorem is the starting point of the theory of differentiation of real functions in measure theory. It leads to the differentiation of indefinite Lebesgue integrals and to the fundamental theorem of calculus for absolutely continuous functions. Fubini's theorem on differentiation refines it for series of monotone functions. In contrast with the Cantor function, absolutely continuous functions satisfy $\int_a^b f' = f(b) - f(a)$.