Advanced Analysis on the Real Line [Paperback]

- < f is increasing. The latter part of the book deals with functions of bounded variation and approximately continuous functions. Finally there is an exhaustive chapter on the generalized Cantor sets and Cantor functions. The bibliography is extensive and a great variety of exercises serves to clarify and sometimes extend the results presented in the text.- < f is increasing. The latter part of the book deals with functions of bounded variation and approximately continuous functions. Finally there is an exhaustive chapter on the generalized Cantor sets and Cantor functions. The bibliography is extensive and a great variety of exercises serves to clarify and sometimes extend the results presented in the text.0 Preliminaries.- 0.1 Lebesgue Measure.- 0.2 The Lebesgue Integral.- 0.3 Vitali Covering Theorem.- 0.4 Baire Category Theorem and Baire Class Functions.- 1 Monotone Functions.- 1.1 Continuity Properties.- 1.2 Differentiability Properties.- 1.3 Reconstruction of f from f?.- 1.4 Series of Monotone Functions.- Exercises.- 2 Density and Approximate Continuity.- 2.1 Preliminaries and Definitions.- 2.2 The Lebesgue Density Theorem.- 2.3 Approximate Continuity.- 2.4 Approximate Continuity and Integrability.- 2.5 Further Results on Approximate Continuity.- 2.6 Sierpinskis Theorem.- 2.7 The Darboux Property and the Density Topology.- Exercises.- 3 Dini Derivatives.- 3.1 Preliminaries and Definitions.- 3.2 Simple Properties of Derivatives.- 3.3 Ruziewiczs Example.- 3.4 Further Properties of Derivatives.- 3.5 The Denjoy-Saks-Young Theorem.- 3.6 Measurability of Dini Derivatives.- 3.7 Dini Derivatives and Convex Functions.- Exercises.- 4 Approximate Derivatives.- 4.1 Definitions.- 4.2 Measurability of Approximate Derivatives.- 4.3 Analogue of the Denjoy-Saks-Young Theorem.- 4.4 Category Results for Approximate Derivatives.- 4.5 Other Properties of Approximate Derivatives.- Exercises.- 5 Additional Results on Derivatives.- 5.1 Derivatives.- 5.2 Derivates.- 5.3 ApproxlƒJ