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Title Details:
Analysis
Other Titles: Real Functions of one Variable
Authors: Papadimitrakis, Michail
Reviewer: Sarantopoulos, Ioannis
Subject: MATHEMATICS AND COMPUTER SCIENCE > MATHEMATICS > REAL FUNCTIONS
Keywords:
Limit
Continuity
Integral
Derivative
Sequence
Metric space
Series
Description:
Abstract:
The classical content of Infinitesimal Calculus (mainly of functions of one variable) with an empasis on the rigorous foundations and on the theoretical proofs of theorems. The supremum property. Existence of roots. Rigorous definition of powers with rational and irrational exponents and of logarithms. Sequences (monotonic, Cauchy, the Bolzano-Weierstrass theorem, limsup-liminf). Limits of functions (monotonicity, Cauchy criterion). Continuity of functions and the basic theorems. Continuity of the inverse function. Uniform continuity. Derivative and the basic theorems. Monotonicity. Convexity. Indeterminate forms. Taylor’s theorems. Integral. Methods of Darboux and of Riemann. Properties of the integral. Relation between derivative and integral. Calculation of integrals. Series of numbers. Criteria of convergence. Sequences of functions. Uniform convergence. The theorem of Weierstrass. Series of functions. Uniform convergence. Criteria of convergence. Power series. Trigonometric functions. Metric spaces. Completeness. Compactness. Connectedness. Generalised integrals. Criteria of convergence. Integrals with parameter. Axiomatic foundation. Dedekind cuts.
Type: Undergraduate textbook
Creation Date: 2015
Item Details:
ISBN 978-960-603-403-9
License: http://creativecommons.org/licenses/by-nc-nd/3.0/gr
DOI http://dx.doi.org/10.57713/kallipos-739
Handle http://hdl.handle.net/11419/2890
Bibliographic Reference: Papadimitrakis, M. (2015). Analysis [Undergraduate textbook]. Kallipos, Open Academic Editions. https://dx.doi.org/10.57713/kallipos-739
Language: Greek
Consists of:
1. The real numbers
2. Sequences and limits of sequences
3. Limits of functions
4. Continuous functions
5. Derivatives of functions
6. Riemann integrals
7. Relation between derivative and integral
8. Series of numbers
9. Sequences of functions
10. Series of functions
11. Metric spaces
12. Generalised integrals
13. Axiomatic foundations
14. Hints and solutions of exercises
Number of pages 898
Publication Origin: Kallipos, Open Academic Editions
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