This classic textbook has been used successfully by instructors and students for nearly three decades. This timely new edition offers minimal yet notable changes while retaining all the elements, presentation, and accessible exposition of previous editions. A list of updates is found in the Preface to this edition.

This text is based on the author’s experience in teaching graduate courses and the minimal requirements for successful graduate study. The text is understandable to the typical student enrolled in the course, taking into consideration the variations in abilities, background, and motivation. Chapters one through six have been written to be accessible to the average student,

w hile at the same time challenging the more talented student through the exercises.

Chapters seven through ten assume the students have achieved some level of expertise in the subject. In these chapters, the theorems, examples, and exercises require greater sophistication and mathematical maturity for full understanding.

In addition to the standard topics the text includes topics that are not always included in comparable texts.

• Chapter 6 contains a section on the Riemann-Stieltjes integral and a proof of Lebesgue’s t heorem providing necessary and sufficient conditions for Riemann integrability.
• Chapter 7 also includes a section on square summable sequences and a brief introduction to normed linear spaces.
• C hapter 8 contains a proof of the Weierstrass approximation theorem using the method of

aapproximate identities.

• The inclusion of Fourier series in the text allows the student to gain some exposure to this important subject.
• The final chapter includes a detailed treatment of Lebesgue measure and the Lebesgue integral, using inner and outer measure.
• The exercises at the end of each section reinforce the concepts.
• Notes provide historical comments or discuss additional topics.



1.  The Real Numbers
2.  Topology of the Real Line
3.  Sequences of Real Numbers
   
4.  Limits and Continuity
   
5.  Differentiation
   
6.  Integration
   
7.  Series of Real Numbers
   
8.  Sequences and Series of Functions
   
9.  Fourier Series
   
10.  Lebesgue Measure and Integrations
    

Manfred Stoll received his Ph.D. from Penn State University under the supervision of K.T. Hahn and supported by an NDEA Fellowship. He has spent his entire career at University of South Carolina, serving as Department Chair, and supervised 3 master's students and eight doctoral students. He has published over 47 refereed research articles and published three books. He has served as a Referee on over 22 journals, served on numerous Panel Sessions for the AMS and Review Panels for the Science Foundation of Ireland (2005-2008), and was the Program Officer in Mathematics and Physics for the Science Foundation of Ireland in Summer 2007. Since 1990 he has given 27 invited conference talks, including hour addresses and plenary talks. 

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