Differentiation of Real Functions

Andrew M. Bruckner (auth.)3540089101, 9783540089100, 0387089101

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.

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Table of contents :
Preliminaries….Pages 1-2
Darboux functions….Pages 3-7
Darboux functions in the first class of Baire….Pages 8-44
Continuity and approximate continuity of derivatives….Pages 45-51
The extreme derivates of a function….Pages 52-70
Reconstruction of the primitive….Pages 71-84
The Zahorski classes….Pages 85-97
The problem of characterizing derivatives….Pages 98-110
Derivatives a.e. and generalizations….Pages 111-122
Transformations via homeomorphisms….Pages 123-146
Generalized derivatives….Pages 147-172
Monotonicity….Pages 173-198
Stationary and determining sets….Pages 199-208
Behavior of typical continuous functions….Pages 209-223
Miscellaneous topics….Pages 224-233