Differentiation

This chapter contains first a brief but reasonably complete treatment of the theory of differentiation for complex-valued functions defined on intervals of the line. Section 17 is severely classical, containing examples and Lebesgue’s famous theorem on differentiation of functions of finite variation. In § 18, we explore the conditions under which the classical equality $$f(b) - f(a) = \int\limits_a^b {f'(t) \ dt}$$ is valid. This exploration leads to interesting and perhaps unexpected measure-theoretic ideas, which have little to do with differentiation and which have applications in extraordinarily diverse fields. The main result in this direction is the Lebesgue-Radon-Nikodým theorem, which we examine thoroughly in § 19 and apply to the decomposition of measures on R. In § 20, we present several other applications of the Lebesgue-Radon-Nikodým theorem to problems in abstract analysis. Sections 17 and 18 are important, and should be studied by all readers. The same is true of § 19, up to and including (19.24). The remainder of § 19 may be omitted by readers pressed for time. Of § 20, (20.1)–(20.5) and (20.41)–(20.52) are topics important for every student. The remainder of § 20 is in our opinion interesting but less vital, and it too may be omitted by readers pressed for time.