Absolutely continuous functions

A function $F : [a,b] \to \mathbb{R}$ is absolutely continuous when for every $\varepsilon > 0$ there is $\delta > 0$ such that for every finite family of pairwise disjoint subintervals $(a_1, b_1), \dots, (a_N, b_N)$ of $[a,b]$,

$\sum_{k=1}^{N} (b_k - a_k) < \delta \ \Longrightarrow \ \sum_{k=1}^{N} |F(b_k) - F(a_k)| < \varepsilon.$

Write $AC[a,b]$ for the class of such $F$. The inclusions

$\text{Lipschitz} \subsetneq AC[a,b] \subsetneq \{\text{continuous and of bounded variation}\} \subsetneq \{\text{continuous}\}$

are all strict: $x \mapsto \sqrt{x}$ on $[0,1]$ is absolutely continuous but not Lipschitz; the Cantor function is continuous, increasing and of bounded variation but not absolutely continuous; and $x \mapsto x \sin(1/x)$ on $(0,1]$, extended by $0$, is continuous but not of bounded variation. $AC[a,b]$ is a vector space, closed under products, and $F \in AC[a,b]$ has Luzin's property (N): it maps null sets to null sets.

Partly proved elsewhere in this library. The elementary definition is Absolute continuity on a compact interval ↗. The hierarchy $C^1\subseteq\mathrm{Lipschitz}\subseteq AC\subseteq C\cap BV$ and strictness witnesses are proved in $C^1$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation ↗ and its companion examples. Closure under vector-space operations and products, Luzin's property (N), the sharp Lebesgue-integral FTC and the Banach–Zarecki characterisation remain unproved.

What would prove it. Absolute continuity implies bounded variation by a covering argument on $[a,b]$; property (N) follows from the definition applied to a cover of the null set by intervals of small total length; strictness of the inclusion at the Cantor function needs that $c$ maps a null set onto a set of measure $1$, and that is a genuine measure statement. The real theorem about the class is The sharp fundamental theorem of calculus (absolute continuity) ‡, and its characterisation without integrals is Banach-Zarecki theorem ‡.

Which page it serves. The bounded variation and Riemann-Stieltjes page, which builds the theory of $BV[a,b]$, the Jordan decomposition into a difference of increasing functions, and the total variation function. Absolute continuity is the next class in that hierarchy and the one for which the Newton-Leibniz formula is exactly true, so the page can define it, place it in the hierarchy with the counterexamples above, and then must stop.

What is in scope on that page. The whole $BV$ theory, the Jordan decomposition, the Cantor function as a continuous increasing non-constant function with derivative zero off a null set, and the elementary statement that absolutely continuous implies uniformly continuous and of bounded variation. What is not in scope is any statement whose formulation needs the Lebesgue integral.