Differentiation

Abstract

Integrable and step functions (the latter are “dense” among the former). Covering of a set by a family of closed intervals such that every number in the set belongs to an arbitrarily short interval of the family (“Vitali covering”); the main theorem about Vitali coverings. Generalization of derivative by using limsup and liminf instead of limit, called “Dini derivatives”. Differentiability almost everywhere of monotone functions (Lebesgue) via Vitali coverings; derivative of the indefinite integral equals the integrand almost everywhere. Motivation for the concept of total variation. Bounded variation and monotonicity. Absolute continuity; an absolutely continuous function and the indefinite integral of its derivative. Example of a continuous and strictly increasing function whose value increases by more than the integral of its derivative.