Elements of Mathematics Functions of a Real Variable

Elementary Theory

  • Nicolas Bourbaki
  • Philip Spain

Table of contents

  1. Front Matter
    Pages I-XIV
  2. Elementary Theory, Philip Spain
    Pages 1-2
  3. Elementary Theory, Philip Spain
    Pages 3-49
  4. Elementary Theory, Philip Spain
    Pages 51-90
  5. Elementary Theory, Philip Spain
    Pages 91-162
  6. Elementary Theory, Philip Spain
    Pages 163-209
  7. Elementary Theory, Philip Spain
    Pages 211-267
  8. Elementary Theory, Philip Spain
    Pages 269-303
  9. Elementary Theory, Philip Spain
    Pages 305-331
  10. Back Matter
    Pages 333-338

About this book


This book is an English translation of the last French edition of Bourbaki’s Fonctions d'une Variable Réelle.

The first chapter is devoted to derivatives, Taylor expansions, the finite increments theorem, convex functions. In the second chapter, primitives and integrals (on arbitrary intervals) are studied, as well as their dependence with respect to parameters. Classical functions (exponential, logarithmic, circular and inverse circular) are investigated in the third chapter. The fourth chapter gives a thorough treatment of differential equations (existence and unicity properties of solutions, approximate solutions, dependence on parameters) and of systems of linear differential equations. The local study of functions (comparison relations, asymptotic expansions) is treated in chapter V, with an appendix on Hardy fields. The theory of generalized Taylor expansions and the Euler-MacLaurin formula are presented in the sixth chapter, and applied in the last one to the study of the Gamma function on the real line as well as on the complex plane.

Although the topics of the book are mainly of an advanced undergraduate level, they are presented in the generality needed for more advanced purposes: functions allowed to take values in topological vector spaces, asymptotic expansions are treated on a filtered set equipped with a comparison scale, theorems on the dependence on parameters of differential equations are directly applicable to the study of flows of vector fields on differential manifolds, etc.


MSC (2000): 26xx YellowSale2006 calculus derivatives gamma function integrals real functions Convexity derivative differential equation integral integration logarithm manifold mean value theorem

Authors and affiliations

  • Nicolas Bourbaki
  • Philip Spain
    • 1
  1. 1.Department of MathematicsUniversity of GlasgowGlasgowScotland

Bibliographic information

  • DOI
  • Copyright Information Springer Berlin Heidelberg 2004
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-63932-6
  • Online ISBN 978-3-642-59315-4
  • Buy this book on publisher's site
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