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Problems in Real Analysis by Titu Andreescu, Teodora-Liliana T. Radulescum, Vicentiu D. Radulescu
(Advanced Calculus on the Real Axis)

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Problems in Real Analysis written by Titu Andreescu , School of Natural Sciences and Mathematics, University of Texas at Dallas, Richardson, TX 75080, USA, titu.andreescu@utdallas.edu, Teodora-Liliana T. Radulescum , Department of Mathematics, Fratii Buzesti National College, Craiova 200352, Romania, teodoraradulescu@yahoo.com and Vicentiu D. Radulescu , Simion Stoilow Mathematics Institute, Romanian Academy, Bucharest 014700, Romania, vicentiu.radulescu@math.cnrs.fr.
The book is mainly geared toward students studying the basic principles of mathematical analysis. However, given its selection of problems, organization, and level, it would be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam and other high-level mathematical contests. We also address this work to motivated high-school and undergraduate students. This volume is meant primarily for students in mathematics, physics, engineering, and computer science, but, not without authorial ambition, we believe it can be used by anyone who wants to learn elementary mathematical analysis by solving problems. The book is also a must-have for instructors wishing to enrich their teaching with some carefully chosen problems and for individuals who are interested in solving difficult problems in mathematical analysis on the real axis. The volume is intended as a challenge to involve students as active participants in the course. To make our work self-contained, all chapters include basic definitions and properties. The problems are clustered by topic into eight chapters, each of them containing both sections of proposed problems with complete solutions and separate sections including auxiliary problems, their solutions being left to our readers. Throughout the book, students are encouraged to express their own ideas, solutions, generalizations, conjectures, and conclusions.
The volume contains a comprehensive collection of challenging problems, our goal being twofold: first, to encourage the readers to move away from routine exercises and memorized algorithms toward creative solutions and nonstandard problem-solving techniques; and second, to help our readers to develop a host of new mathematical tools and strategies that will be useful beyond the classroom and in a number of applied disciplines. We include representative problems proposed at various national or international competitions, problems selected from prestigious mathematical journals, but also some original problems published in leading publications. That is why most of the problems contained in this book are neither standard nor easy. The readers will find both classical topics of mathematical analysis on the real axis and modern ones. Additionally, historical comments and developments are presented throughout the book in order to stimulate further inquiry.

Problems in Real Analysis written by Titu Andreescu, Teodora-Liliana T. Radulescum, Vicentiu D. Radulescu cover the following topics.

  • Part I Sequences, Series, and Limits
    1. Sequences
    2. Series
    3. Limits of Functions

  • Part II Qualitative Properties of Continuous and Differentiable Functions
    4. Continuity
    5. Differentiability

  • Part III Applications to Convex Functions and Optimization
    6. Convex Functions
    7. Inequalities and Extremum Problems

  • Part IV Antiderivatives, Riemann Integrability, and Applications
    8. Antiderivatives
    9. Riemann Integrability
    10. Applications of the Integral Calculus

  • Part V Appendix
    A. Basic Elements of Set Theory
    A.1 Direct and Inverse Image of a Set
    A.2 Finite, Countable, and Uncountable Sets
    B. Topology of the Real Line
    B.1 Open and Closed Sets
    B.2 Some Distinguished Points

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