It is well known that the fundamental theorem of calculus is essentially characterised by absolutely continuous functions. In fact, it fails drastically when we look at the class of bounded variation functions, e.g. the Cantor stair function increases from 0 to 1 however has derivative zero almost everywhere. When learning measure theory as an undergraduate, I was shown a fundamental theorem of calculus, attributed to de la Vallée Poussin, which accounts for these defects in the usual fundamental theorem of calculus for monotone functions:
Theorem. (de la Vallée Poussin) Let $f: \mathbb R \to \mathbb R$ be non-decreasing and continuous, and denote $\mu$ its associated Lebesgue-Stieltjes measure. Then $f$ is differentiable Lebesgue-a.e. and $\mu$-a.e., with $|\{f' = \infty\}| = 0$ and \begin{equation} f(b) - f(a)= \int_{(a, b] \cap \{f' < \infty\}} f' \, dx + \mu((a, b] \cap \{f' = \infty \}). \end{equation}
Showing that the integral of the derivative is controlled by the left-hand side is a routine textbook result, c.f. for example Stein-Shakarchi. However, it would be greatly appreciated if someone could point me to a reference to the result above, such as the original paper or a textbook, as I have not been able to locate one myself (apart from old lecture notes!)