Advanced Analysis: on the Real Line

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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.