The goal of this book is to provide an extensive collection of results which generalize classical real analysis. Besides discussing density, approximate continuity, and approximate derivatives in detail, culminating with the Denjoy-Saks-Young Theorem, the authors also present an interesting example due to Ruziewicz on an infinite number of functions with the same derivative (not everywhere finite) but the difference of any two is not a constant and Sierpinski's theorem on the extension of approximate continuity to nonmeasurable functions. There is also a chapter on monotonic functions and one dealing with the Tonelli-Goldowsky result of the weakening of the hypotheses on a function f such that f'r > - < 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.

Table of Contents:
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 Sierpinski’s 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 Ruziewicz’s 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 Approximate Derivatives.- 5.4 The Denjoy Property.- 5.5 Metrically Dense.- 6 Bounded Variation.- 6.1 Bounded Variation of Finite Intervals.- 6.2 Stieltjes Integral.- 6.3 The Space BV[a,b].- BVloc and L1loc.- 6.5 Additional Remarks on Fubini’s Theorem.- Exercises.- 7 Absolute Continuity.- 7.1 Absolute Continuity.- 7.2 Rectifiable Curves.- Exercises.- 8 Cantor Sets and Singular Functions.- 8.1 The Cantor Ternary Set and Function.- 8.2 Hausdorff Measure.- 8.3 Generalized Cantor Sets—Part I.- 8.4 Generalized CantorSets—Part II.- 8.5 Cantor-like Sets.- 8.6 Strictly Increasing Singular Functions.- Exercises.- 9 Spaces of BV and AC Functions.- 9.1 Convergence in Variation.- 9.2 Convergence in Length.- 9.3 Norms on AC.- 9.4 Norms on BV.- 10 Metric Separability.- Exercises.