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Strange Functions in Real Analysis, Third Edition
"Strange Functions in Real Analysis, Third Edition differs from the previous editions in that it includes five new chapters as well as two appendices. More importantly, the entire text has been revised and contains more detailed explanation's of the presented material. In doing so, the book explores a number of important examples and constructions of pathological functions. After introducing basic concepts, the author begins with Cantor and Peano-type functions, then moves effortlessly to functions whose constructions require what is essentially non-effective methods. These include functions without the Baire property, functions associated with a Hamel basis of the real line and Sierpinski-Zygmund functions that are discontinuous on each subset of the real line having the cardinality continuum.Finally, the author considers examples of functions whose existence cannot be established without the help of additional set-theoretical axioms. On the whole, the book is devoted to strange functions (and point sets) in real analysis and their applications. "--Provided by publisher.
chapter 0 Introduction: Basic concepts -- chapter 1 Cantor and Peano type functions -- chapter 2 Functions of first Baire class -- chapter 3 Semicontinuous functions that are not countably continuous -- chapter 4 Singular monotone functions -- chapter 5 A characterization of constant functions via Dini's derived numbers -- chapter 6 Everywhere differentiable nowhere monotone functions -- chapter 7 Continuous nowhere approximately differentiable functions -- chapter 8 Blumberg's theorem and Sierpin´ski-Zygmund functions -- chapter 9 The cardinality of first Baire class -- chapter 10 Lebesgue nonmeasurable functions and functions without the Baire property -- chapter 11 Hamel basis and Cauchy functional equation -- chapter 12 Summation methods and Lebesgue nonmeasurable functions -- chapter 13 Luzin sets, Sierpin´ski sets, and their applications -- chapter 14 Absolutely nonmeasurable additive functions -- chapter 15 Egorov type theorems -- chapter 16 A difference between the Riemann and Lebesgue iterated integrals -- chapter 17 Sierpin´ski's partition of the Euclidean plane -- chapter 18 Bad functions defined on second category sets -- chapter 19 Sup-measurable and weakly sup-measurable functions -- chapter 20 Generalized step-functions and superposition operators -- chapter 21 Ordinary differential equations with bad right-hand sides -- chapter 22 Nondifferentiable functions from the point of view of category and measure -- chapter 23 Absolute null subsets of the plane with bad orthogonal projections.