โ Scribed by Alexander Kharazishvili
No coin nor oath required. For personal study only.
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 explanations 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.
Content: Introduction: basic concepts Cantor and Peano type functions Functions of first Baire class Semicontinuous functions that are not countably continuous Singular monotone functions A characterization of constant functions via Dini's derived numbers Everywhere differentiable nowhere monotone functions Continuous nowhere approximately differentiable functions Blumberg's theorem and Sierpinski-Zygmund functionsThe cardinality of first Baire classLebesgue nonmeasurable functions and functions without the Baire property Hamel basis and Cauchy functional equation Summation methods and Lebesgue nonmeasurable functions Luzin sets, Sierpi'nski sets, and their applicationsAbsolutely nonmeasurable additive functions Egorov type theorems A difference between the Riemann and Lebesgue iterated integrals Sierpinski's partition of the Euclidean plane Bad functions defined on second category sets Sup-measurable and weakly sup-measurable functions Generalized step-functions and superposition operators Ordinary differential equations with bad right-hand sides Nondifferentiable functions from the point of view of category and measure Absolute null subsets of the plane with bad orthogonal projectionsAppendix 1: Luzin's theorem on the existence of primitives Appendix 2: Banach limits on the real line
Functional analysis.;Functions of real variables.
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Analyzes examples and constructions of strange functions. Explores the Axiom of Dependent Choice and demonstrates its sufficiency for most domains of classical mathematics. Highlights the general theory of stochastic processes. Contains more than 1400 equations.