Real Analysis – vol 1 – Functions of One Variable
Authors
Description
Real Analysis – Functions of One Variable is an introduction to the study of real functions of one real variable, aimed at university students who already have experience equivalent to one or two semesters of Calculus.
The presentation is elementary, with illustrative examples. Practically all the proposed exercises are solved in the final chapter. The author has taken care to carefully justify all the statements made, coherently but without formal exaggeration. The choice of topics has aimed to strike a balance between usefulness and possible applications.
Target audience
Higher education
Name: Real Analysis – vol 1 – Functions of One Variable
Author(s): e Elon Lages Lima
Pages: 216
Publication: IMPA, 2020
ISBN: 978-65-990528-5-9
Edition: 13
1 Finite and Infinite Sets
1 Natural numbers
2 Finite sets
3 Infinite sets
4 Enumerable sets
5 Exercises
2 Real Numbers
1 R is a body
2 R is an ordered body
3 R is a complete ordered body
4 Exercises
3 Sequences of Real Numbers
1 Limit of a sequence
2 Limits and inequalities
3 Operations with limits
4 Infinite limits
5 Exercises
4 Numerical series
1 Convergent series
2 Absolutely convergent series
3 Convergence tests
4 Commutativity
5 Exercises
5 Some Topological Notions
1 Open sets
2 Closed sets
3 Accumulation points
4 Compact sets
5 The Cantor set
6 Exercises
6 Limits of Functions
1 Definition and first properties
2 Lateral limits
3 Limits at infinity, infinite limits, indeterminate expressions
4 Exercises
7 Continuous functions
1 Definition and first properties
2 Continuous functions on an interval
3 Continuous functions on compact sets
4 Uniform continuity
5 Exercises
8 Derivatives
1 The notion of derivative
2 Operational rules
3 Derivative and local growth
4 Derivable functions on an interval
5 Exercises
9 Taylor’s formula and applications of the derivative
1 Taylor’s formula
2 Convex and concave functions
3 Successive approximations and Newton’s method
4 Exercises
10 The Riemann Integral
1 Review of sup and inf
2 The Riemann Integral
3 Properties of the integral
4 Integrability conditions
5 Exercises
11 Calculus with Integrals
1 The classic theorems of Integral Calculus
2 The integral as the limit of Riemann sums
3 Logarithms and exponentials
4 Improper integrals
5 Exercises
12 Sequences and Series of Functions
1 Simple convergence and uniform convergence
2 Properties of uniform convergence
3 Power series
4 Trigonometric functions
5 Taylor series
6 Exercises
13 Suggestions and Answers
1 Finite and Infinite Sets
2 Real Numbers
3 Sequences of Real Numbers
4 Numerical Series
5 Some Topological Notions
6 Limits of Functions
7 Continuous Functions
8 Derivatives
9 Taylor’s Formula and Applications of the Derivative
10 The Riemann Integral
11 Calculus with Integrals
12 Sequences and Series of Functions
Bibliography
