Real Functions
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1170 Brian S. Thomson
Real Functions
Springer-Verlag Berlin Heidelberg New York Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
1170 Brian S. Thomson
Real Functions
Springer-Verlag Berlin Heidelberg New York Tokyo
Author
Brian S. Thomson Department of Mathematics and Statistics, Simon Fraser University British Columbia, Canada, V5A 1S6
Mathematics Subject Classification (1980): 26-02 ISBN 3-540-16058-2 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-16058-2 Springer-Verlag New York Heidelberg Berlin Tokyo
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PREFACE
Our intention in this mongraph is to survey a number of topics related to the study of the continuity and differentiation properties of real functions, in certain generalized senses. There is now a relatively large literature devoted to such subtle concerns but which is accessible and known only to specialists.
Since the ideas are essentially simple and the techniques required
fairly elementary this literature should be easily absorbed by any interested mathematician, and it is hoped that the presentation here is sufficiently readable and the exposition adequately clear for this purpose. Probably the reader needs only a famil iarity with the usual basics of real analysis (measure, category, density, erc.) in order to follow the arguments.
This material is readily available in a
variety of textbooks. A better preparation would be to master the books SvSaks , Theory of the integral,
and A.M.Bruckner, Differentiation of real functions, (references [33] and [209] in the bibliography) that most analysts who work in this particular set of topics would surely consider fundamental to our subject. The present monograph continues certain concerns that arise in each of these works. Part of this material was presented in a series of seminars at the University of California at Santa Barbara in the spring of 1984, during the special year in Real Analysis that was held there. I am particularly grateful and certainly indebted to the participants in that seminar who offered much helpful criticism and indicated numerous improvements. What remains is, doubtless, flawed but much less so than it would have been without the opportunity to meet with so many fine analysts. In the first chapter is presented a general structure (called here a local system of sets) that can be used to formulate a variety of general notions of limit, continuity, derivative, etc. for real functions.
The reason we have chosen this abstrac
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