This book explores a number of important examples and constructions of pathological functions. After introducing the basic concepts, the author begins with Cantor and Peano-type functions, then moves to functions whose constructions require essentially noneffective methods.
This book explores a number of important examples and constructions of pathological functions. After introducing the basic concepts, the author begins with Cantor and Peano-type functions, then moves to functions whose constructions require essentially noneffective methods.
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 functions The cardinality of first Baire class Lebesgue 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 applications Absolutely 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 projections Appendix 1: Luzin's theorem on the existence of primitives Appendix 2: Banach limits on the real line
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 functions The cardinality of first Baire class Lebesgue 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 applications Absolutely 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 projections Appendix 1: Luzin's theorem on the existence of primitives Appendix 2: Banach limits on the real line
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