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.
Author(s): Andrew Bruckner
Series: CRM Monograph Series 5
Edition: 2 Sub
Publisher: American Mathematical Society
Year: 1994
Language: English
Pages: C+xii, 195, B
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
