Preface I - Sets and Functions §1. Set Theory 1 - Membership, equality, empty set 2 - The set defined by a relation. Intersections and unions 3 - Whole numbers. Infinite sets 4 - Ordered pairs, Cartesian products, sets of subsets 5 - Functions, maps, correspondences 6 - Injections, surjections, bijections 7 - Equipotent sets. Countable sets 8 - The different types of infinity 9 - Ordinals and cardinals §2. The logic of logicians II - Convergence: Discrete variables §1. Convergent sequences and series 0 - Introduction: what is a real number? 1 - Algebraic operations and the order relation: axioms of R 2 - Inequalities and intervals 3 - Local or asymptotic properties 4 - The concept of limit. Continuity and differentiability 5 - Convergent sequences: definition and examples 6 - The language of series 7 - The marvels of the harmonic series 8 - Algebraic operations on limits §2. Absolutely convergent series 9 - Increasing sequences. Upper bound of a set of real number 10 - The function log x. Roots of a positive number 11 - What is an integral? 12 - Series with positive terms 13 - Alternating series 14 - Classical absolutely convergent series 15 - Unconditional convergence: general case 16 - Comparison relations. Criteria of Cauchy and d'Alembert 17 - Infinite limits 18 - Unconditional convergence: associativity §3. First concepts of analytic functions 19 - The Taylor series 20 - The principle of analytic continuation 21 - The function cot x and the series ∑ 1/n2k 22 - Multiplication of series. Composition of analytic functions. Formal series 23 - The elliptic functions of Weierstrass III- Convergence: Continuous variables §1. The intermediate value theorem 1 - Limit values of a function. Open and closed sets 2 - Continuous functions 3 - Right and left limits of a monotone function 4 - The intermediate value theorem §2. Uniform convergence 5 - Limits of continuous functions 6 - A slip up of Cauchy's 7 - The uniform metric 8 - Series of continuous functions. Normal convergence §3. Bolzano-Weierstrass and Cauchy's criterion 9 - Nested intervals, Bolzano-Weierstrass, compact sets 10 - Cauchy's general convergence criterion 11 - Cauchy's criterion for series: examples 12 - Limits of limits 13 - Passing to the limit in a series of functions §4. Differentiable functions 14 - Derivatives of a function 15 - Rules for calculating derivatives 16 - The mean value theorem 17 - Sequences and series of differentiable functions 18 - Extensions to unconditional convergence §5. Differentiable functions of several variables 19 - Partial derivatives and differentials 20 - Differentiability of functions of class C1 21 - Differentiation of composite functions 22 - Limits of differentiable functions 23 - Interchanging the order of differentiation 24 - Implicit functions Appendix to Chapter III 1 - Cartesian spaces and general metric spaces 2 - Open and closed sets 3 - Limits and Cauchy's criterion in a metric space; complete spaces 4 - Continuous functions 5 - Absolutely convergent series in a Banach space 6 - Continuous linear maps 7 - Compact spaces 8 - Topological spaces IV - Powers, Exponentials, Logarithms, Trigonometric Functions §1. Direct construction 1 - Rational exponents 2 - Definition of real powers 3 - The calculus of real exponents 4 - Logarithms to base a. Power functions 5 - Asymptotic behaviour 6 - Characterisations of the exponential, power and logarithmic functions 7 - Derivatives of the exponential functions: direct method 8 - Derivatives of exponential functions, powers and logarithms §2. Series expansions 9 - The number e. Napierian logarithms 10 - Exponential and logarithmic series: direct method 11 - Newton's binomial series 12 - The power series for the logarithm 13 - The exponential function as a limit 14 - Imaginary exponentials and trigonometric functions 15 - Euler's relation chez Euler 16 - Hyperbolic functions §3. Infinite products 17 - Absolutely convergent infinite products 18 - The infinite product for the sine function 19 - Expansion of an infinite product in series 20 - Strange identities §4. The topology of the functions Arg(z) and Log z Index
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