Limits and Continuity – Korovkin

In this post, we will see the book Limits and Continuity by P. P. Korovkin.

About the book

The present volume in The Pocket Mathematical Library stems in large part from Chapters 1-4 of P. P. Korovkin’s Mathema­tical Analysis, Moscow (1963). The material has been heavily
rewritten and supplemented by 21 problem sets, one after each section. The result is a succinct but remarkably complete introduction to the theory o f limits and continuity. The book may also be thought of as a “precalculus” text in that it deals with those properties of functions which can be successfully discussed short of introducing the notion of a derivative.

Answers to the even-numbered problems will be found at the end of the book.

The book was translated from Russian by was published in 1969. The book is a part of the Pocket Mathematical Library Series.

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You can get the book here.

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Chapter 1. Functions 1

1. Variables and Functions. Intervals and Sequences 1
2. Absolute Values. Neighborhoods 9
3. Graphs and Tables 12
4. Some Simple Function Classes 14
5. Real Numbers and Decimal Expansions 21

Chapter 2. Limits 24

6. Basic Concepts 24
7. Algebraic Properties of Limits 32
8. Limits Relative to a Set. One-Sided Limits 37
9. Infinite Limits. Indeterminate Forms 43
10. Limits at Infinity 49
11. Limits of Sequences. The Greatest Lower Bound Property 53
12. The Bolzano-Weierstrass Theorem. The Cauchy Convergence Criterion 58
13. Limits of Monotonic Functions. The Function a^{x} and
the Number e 62

Chapter 3. Continuity 73

14. Continuous Functions 72
15. One-Sided Continuity. Classification of Discontinuities 79
16. The Intermediate Value Theorem. Absolute Extrema 83
17. Inverse Functions 89
18. Elementary Functions 92
19. Evaluation of Limits 101
20. Asymptotes 105
21. The Modulus of Continuity. Uniform Continuity 111

Answers to Even-Numbered Problems 116
Index 123

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