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Differentiation of Real Functions
Andrew Bruckner, University of California, Santa Barbara, CA
A co-publication of the AMS and Centre de Recherches Mathématiques.
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CRM Monograph Series
1994; 195 pp; hardcover
Volume: 5
ISBN-10: 0-8218-6990-6
ISBN-13: 978-0-8218-6990-1
List Price: US$65
Member Price: US$52
Order Code: CRMM/5
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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.

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.

Table of Contents

  • Darboux functions
  • Darboux functions in the first class of Baire
  • Continuity and approximate continuity of derivatives
  • The extreme derivatives of a function
  • Reconstruction of the primitive
  • The Zahorski classes
  • The problem of characterizing derivatives
  • Derivatives a.e. and generalizations
  • Transformations via homeomorphisms
  • Generalized derivatives
  • Monotonicity
  • Stationary and determining sets
  • Behavior of typical continuous functions
  • Miscellaneous topics
  • Recent developments
  • Bibliography
  • Supplementary bibliography
  • Terminology index
  • Notational index
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