Introduction to Infinitesimal Analysis: Functions of One Real Variable

Transcriber's Notes.......Page 3
Title Page.......Page 5
Rational and Irrational Numbers.......Page 11
Axiom of Continuity.......Page 12
Addition and Multiplication of Irrationals.......Page 16
General Remarks on the Number System.......Page 18
Axioms for the Real Number System.......Page 19
The Number e.......Page 21
The Transcendence of e.......Page 24
The Transcendence of .......Page 28
Correspondence of Numbers and Points.......Page 33
Segments and Intervals. Theorem of Borel.......Page 34
Limit Points. Theorem of Weierstrass.......Page 38
Second Proof of Theorem thm1515.......Page 41
Definition of a Function.......Page 43
Bounded Functions.......Page 45
Monotonic Functions; Inverse Functions.......Page 46
Rational, Exponential, and Logarithmic Functions.......Page 51
Definitions. Limits of Monotonic Functions.......Page 57
The Existence of Limits.......Page 61
Application to Infinite Series.......Page 65
Infinitesimals. Computation of Limits.......Page 68
Further Theorems on Limits.......Page 74
Bounds of Indetermination. Oscillation.......Page 75
Continuity at a Point.......Page 79
Continuity of a Function on an Interval.......Page 80
Functions Continuous on an Everywhere Dense Set.......Page 84
The Exponential Function.......Page 86
The Order of a Function at a Point.......Page 91
The Limit of a Quotient.......Page 94
Indeterminate Forms......Page 96
Rank of Infinitesimals and Infinites.......Page 101
Definition and Illustration of Derivatives.......Page 103
Formulas of Differentiation.......Page 105
Differential Notations.......Page 112
Mean-value Theorems.......Page 114
Taylor's Series.......Page 117
Indeterminate Forms.......Page 121
General Theorems on Derivatives.......Page 125
Definition of the Definite Integral.......Page 131
Integrability of Functions.......Page 134
Computation of Definite Integrals.......Page 138
Elementary Properties of Definite Integrals.......Page 142
The Definite Integral as a Function of the Limits of Integration.......Page 148
Integration by Parts and by Substitution.......Page 151
General Conditions for Integrability.......Page 153
The Improper Definite Integral on a Finite Interval.......Page 163
The Definite Integral on an Infinite Interval.......Page 171
Properties of the Simple Improper Definite Integral.......Page 174
A More General Improper Integral.......Page 178
Existence of Improper Definite Integrals on a Finite Interval......Page 184
Existence of Improper Definite Integrals on the Infinite Interval......Page 188
INDEX.......Page 191