تفاضل‌گیری توابع حقیقی (مجموعهٔ مونوگرافی کرم، جلد ۵)

Topics related to the differentiation of real functions have received considerable attention during the last few decades. This book provides an efficient account of the present state of the subject. Bruckner addresses in detail the problems that arise when dealing with the class Δ′ of derivatives, a class that is difficult to handle for a number of reasons. Several generalized forms of differentiation have assumed importance in the solution of various problems. Some generalized derivatives are excellent substitutes for the ordinary derivative when the latter is not known to exist; others are not. Bruckner studies generalized derivatives and indicates "geometric" conditions that determine whether or not a generalized derivative will be a good substitute for the ordinary derivative. There are a number of classes of functions closely linked to differentiation theory, and these are examined in some detail. The book unifies many important results from the literature as well as some results not previously published. The first edition of this book, which was current through 1976, has been referenced by most researchers in this subject. This second edition contains a new chapter dealing with most of the important advances between 1976 and 1993. Titles in this series are co-published with the Centre de Recherches Mathématiques. Readership: Graduate students and researchers in the differentiation theory of real functions and related subjects. Cover Titles in this Series Differentiation of Real Functions Copyright (C) Copyright 1994 by the American Mathematical Society ISBN 0821869906 QA304.B78 1994 5151 3---dc20 Table of Contents Preface to the Second Edition Preface Introduction Preliminaries CHAPTER 1 Darboux Functions 1. Examples of Darboux functions 2. Remarks 3. Darboux functions and continuity 4. Operations; combinations; and approximations 5. Additional remarks CHAPTER 2 Darboux Functions in the First Class of Baire 1. Equivalences 2. Examples 3. Operations; Combinations and Approximations 4. The class of derivatives: preliminary comparisons with DB1 5. Approximate continuity 6. The Luzin-Menchoff Theorem and constructions of approximately continuous functions 7. Maximoff's Theorems 8. Integral comparisons of C; Cap' tl.; and 'DB1 9. Remarks CHAPTER 3 Continuity and Approximate Continuity of Derivatives 1. Examples of discontinuous derivatives 2. Characterization of the set of discontinuities of a derivative 3. Approximate continuity of the derivative 4. A relationship between Cap and Ll' CHAPTER 4 The Extreme Derivates of a Function 1. Definitions and basic properties 2. Measurability and Baire classifications of extreme derivates 3. A Darboux-like property of Dini derivatives 4. Relationships Among the Derivates CHAPTER 5 Reconstruction of the Primitive 1. Reconstructions by Riemann or Lebesgue integration 2. Reconstruction of the primitive when its derivative is finite 3. Ambiguities when derivatives can be infinite 4. Generalized bounded variation and generalized absolute continuity CHAPTER 6 The Zahorski Classes 1. Definitions and basic properties 2. Derivatives and the classes 3. Related conditions CHAPTER 7 The Problem of Characterizing Derivatives 1. Associated sets 2. Perfect systems 3. An analogue to characterizing integrals 4. A characterization of \Delta ' 5. Miscellaneous remarks CHAPTER 8 Derivatives a.e. and Generalizations 1. Derivatives a.e. 2. A generalized derivative 3. Universal generalized antiderivatives 4. Differentiability a.e. CHAPTER 9 Transformations via Homeomorphisms 1. DifFerentiability via inner homeomorphisms 2. Differentiability via outer homeomorphisms 3. Derivatives via inner homeomorphisms 4. Derivatives via outer homeomorphisms 5. Summary and miscellaneous remarks CHAPTER 10 Generalized Derivatives 1. The approximate derivative--basic properties 2. Behavior of approximate derivatives 3. Miscellany 4. Other generalized derivatives CHAPTER 11 Monotonicity 1. Some historical background for Section 2 2. A general theorem 3. Applications of Theorem 2.5 4. Monotonicity conditions in terms of extreme derivates 5. Monotonicity when D+ F in B1 6. Convexity CHAPTER 12 Stationary and Determining Sets 1. The stationary and determining sets for certain classes 2. Miscellaneous remarks CHAPTER 13 Behavior of Typical Continuous Functions 1. Preliminaries and basic terminology 2. Differentiability structure of typical continuous functions 3. Horizontal level sets 4. Total level set structure 5. Miscellaneous Comments CHAPTER 14 Miscellaneous Topics 1. Restrictive differentiability properties of functions 2. Extensions to derivatives 3. The set of points of differentiability of a function 4. Derivatives, approximate continuity, and summability 5. Additional topics CHAPTER 15 Recent Developments 1. Path derivatives 2. The algebra generated by D.' 3. More about typical behavior 3.1 Porosity considerations 3.2 Besicovitch functions 3.3 Typical behavior in other classes 4. Miscellany Bibliography Supplementary Bibliography Terminology Index Notational Index Topics related to the differentiation of real functions have received considerable attention during the last few decades. This book provides an efficient account of the present state of the subject. Bruckner addresses in detail the problems that arise when dealing with the class $\Delta '$ of derivatives, a class that is difficult to handle for a number of reasons. Several generalized forms of differentiation have assumed importance in the solution of various problems. Some generalized derivatives are excellent substitutes for the ordinary derivative when the latter is not known to exist; others are not. Bruckner studies generalized derivatives and indicates ``geometric'' conditions that determine whether or not a generalized derivative will be a good substitute for the ordinary derivative. There are a number of classes of functions closely linked to differentiation theory, and these are examined in some detail. The book unifies many important results from the literature as well as some results not previously published. The first edition of this book, which was current through 1976, has been referenced by most researchers in this subject. This second edition contains a new chapter dealing with most of the important advances between 1976 and 1993. Topics related to the differentiation of real functions have received considerable attention during the last few decades. This book provides an account of the state of the subject. It addresses the problems that arise when dealing with the class $\Delta '$ of derivatives, a class that is difficult to handle for a number of reasons. Andrew Bruckner. Includes Bibliographical References (p. 181-191) And Indexes.