Differentiation of Real Functions
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659 Andrew M. Bruckner
Differentiation of Real Functions
Springer-Verlag Berlin Heidelberg New York 1978
Author Andrew M. Bruckner Department of Mathematics University of California Santa Barbara, CA 93106/USA
AMS Subject Classifications (1970): primary: 26A24 secondary: 26A21, 26A27, 26A30, 26AS9, 26A45, 26A48, 26A51 ISBN 3-540-08910-1 ISBN 0-387-08910-1
Springer-Verlag Berlin Heidelberg New York Springer-Verlag New York Heidelberg Berlin
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© by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-5432ID
PREFACE It has now been about forty years since the publication of Saks' book, Theory
of the Integral, a book which deals considerably with topics which are related to differentiation theory.
Since that time, particularly since the publication of
Zahorski's paper [216] in 1950, much work has been done related to the differentiation of real functions, but little of it has appeared in book form. A consequence of this is that many results have been reproved or rediscovered several times.
In addition, there are many instances of an author proving a theorem
"from scratch, If that is, without the knowledge of related results which existed at the time, and which could have been used to prove the theorem much more simply. It therefore seems desirable to have a book which (1)
provides a relatively efficient development of the present state of knowledge on the subject,
(2)
discusses some of the open problems which are worth investigating and
(3)
provides references to work on topics which the book does not develop in detail.
These are the main purposes of the present Notes.
It is an outgrowth of courses
and seminars which we have given from timetotime during the last fifteen years at the University of California, Santa Barbara. In order to keep this work to manageable proportions, we have had to make certain compromises.
We tend to omit proofs of those theorems which are either readily
accessible in standard books or which are peripheral to our work.
On occasion,
when several theorems have similar proofs, we prove only one or two of these theorems. Where we do not give a complete proof, however, we provide references. In putting this book together, we have benefited from discussions with many students and colleagues.
Particular thanks are due to Steve Agronsky, Robert Biskner
and Donald Hancock who have carefully read the entire manuscript and made helpf"ul suggestions.
We also wish to thank Ms. Sonia Ospina who typed the manuscript quickly
and effic
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