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# Introductory Analysis

## 2nd Edition

**The Theory of Calculus**

**The Theory of Calculus**

@bul:* Bases most of the various limit concepts on sequential limits, which is done first
* Defines function limits by first developing the notion of continuity (with a sequential limit characterization)
* Contains a thorough development of the Riemann integral, improper integrals (including sections on the gamma function and the Laplace transform), and the Stieltjes integral
* Presents general metric space topology in juxtaposition with Euclidean spaces to ease the transition from the concrete setting to the abstract
@introbul:New to This Edition
@bul:* Contains new Exercises throughout
* Provides a simple definition of subsequence
* Contains more information on function limits and L'Hospital's Rule
* Provides clearer proofs about rational numbers and the integrals of Riemann and Stieltjes
* presents an appendix lists all mathematicians named in the text
* Gives a glossary of symbols

Researchers, professionals, the general public, and librarians who want to expand or enhance their knowledge of calculus limit theories.

Introduction: Mathematical Statements and Proofs * Types of Mathematical Statements * The Structure of Proofs Ordering of the Real Numbers * The Order Axiom * Least Upper Bounds * The Density of the Rational Numbers * Sequence Limits * Convergent Sequences * Algebraic Combinations of Sequences * Infinite Limits * Subsequences and Limit Points * Monotonic Sequences * Completeness of the Real Numbers * The Bolzano--Weierstrass Theorem * Cauchy Sequences * The Nested Intervals Theorem * The Heine--Borel Covering Theorem * Continuous Functions * Continuity * The Sequential Criterion for Continuity * Combinations of Continuous Functions * One-Sided Continuity * Function Limits * The Sequential Criterion for Function Limits * Variations of Function Limits * Consequences of Continuity * The Range of a Continuous Function * The Intermediate Value Property * Uniform Continuity * The Sequential Criterion for Uniform * Continuity * The Derivative * Difference Quotients * The Chain Rule * The Law of the Mean * Cauchy Law of the Mean * Taylor's Formula with Remainder * L'Hopital's Rule * The Riemann Integral * Riemann Sums and Integrable Functions * Basic Properties * The Darboux Criterion for Integrability * Integrability of Continuous Functions * Products of Integrable Functions * The Fundamental Theorem of Calculus * Improper Integrals * Types of Improper Integrals * Integrals over Unbounded Domains * Integrals of Unbounded Functions * The Gamma Function * The Laplace Transform * Infinite Series * Convergent and Divergent Series * Comparison Tests * The Cauchy Condensation Test * Elementary Tests * Delicate Tests * Absolute and Conditional Convergence * Regrouping and Rearranging Series * Multiplication of Series * The Riemann--Stieltjes Integral * Functions of Bounded Variation * The Total Variation Function * Riemann--Stieltjes Sums and Integrals * Integration by Parts * Integrability of Continuous Functions Function Sequences * Pointwise Convergence * Uniform Convergence *

- No. of pages:
- 335

- Language:
- English

- Copyright:
- © 2000

- Published:
- 10th January 2000

- Imprint:
- Academic Press

- Electronic ISBN:
- 9780080918273

- Print ISBN:
- 9780122676550