© 2017

Essential Real Analysis


  • Contains more than 570 exercises of varying difficulty

  • Provides proofs of basic results on existence and regularity of solutions of ordinary differential equations

  • Includes a full treatment of the inverse function theorem in several variables

  • Emphasizes the importance of estimates and computation in analysis


Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Michael Field
    Pages 1-29
  3. Michael Field
    Pages 91-127
  4. Michael Field
    Pages 129-159
  5. Michael Field
    Pages 161-210
  6. Michael Field
    Pages 245-328
  7. Michael Field
    Pages 329-347
  8. Michael Field
    Pages 349-442
  9. Back Matter
    Pages 443-450

About this book


This book provides a rigorous introduction to the techniques and results of real analysis, metric spaces and multivariate differentiation, suitable for undergraduate courses.

Starting from the very foundations of analysis, it offers a complete first course in real analysis, including topics rarely found in such detail in an undergraduate textbook such as the construction of non-analytic smooth functions, applications of the Euler-Maclaurin formula to estimates, and fractal geometry.  Drawing on the author’s extensive teaching and research experience, the exposition is guided by carefully chosen examples and counter-examples, with the emphasis placed on the key ideas underlying the theory. Much of the content is informed by its applicability: Fourier analysis is developed to the point where it can be rigorously applied to partial differential equations or computation, and the theory of metric spaces includes applications to ordinary differential equations and fractals.

Essential Real Analysis will appeal to students in pure and applied mathematics, as well as scientists looking to acquire a firm footing in mathematical analysis. Numerous exercises of varying difficulty, including some suitable for group work or class discussion, make this book suitable for self-study as well as lecture courses.


use of euler maclaurin formula Euler-Maclaurin formula metric space theory metric space in real analysis fourier series power series infinite series and products smooth and analytic functions uniform approximation existence theorem for ODEs derivative of vector-valued map implicit function theorem smooth functions theory real analysis techniques real analysis books 26-01, 40-01, 26Axx, 26Bxx, 26B05, 26B10, 26Exx, 26E05, 26E10 33Bxx, 33B15, 34A12, 40Axx, 42Axx, 42A10, 54Exx, 54-01, 54E35

Authors and affiliations

  1. 1.Engineering Mathematics Department, Merchant Venturers School of EngineeringBristol UniversityUnited Kingdom

About the authors

Michael Field has held appointments in the UK (Warwick University and Imperial College London), Australia (Sydney University) and the US (the University of Houston and Rice University) and has taught a wide range of courses at undergraduate and graduate level, including real analysis, partial differential equations, dynamical systems, differential manifolds, Lie groups, complex manifolds and sheaf cohomology. His publications in the areas of equivariant dynamical systems and network dynamics include nine books and research monographs as well as many research articles. His computer graphic art work, based on symmetric dynamics, has been widely exhibited and is on display at a number of universities around the world.

Bibliographic information


“This is a well written text on Real Analysis that may be used for a course in Advanced Calculus. It can also serve as a reference for advanced topics in Real Analysis.” (Charles Traina, MAA Reviews, January 4, 2020)

“This book contains a reasonably complete exposition of real analysis which is needed for beginning undergraduate-level students. … This is a well-written textbook with an abundance of worked examples and exercises that are intended for a first course in analysis. This book offers a sound grounding in analysis. In particular, it gives a solid base in real analysis from which progress to more advanced topics may be made.” (Teodora-Liliana Rădulescu, zbMATH 1379.26001, 2018)