This is the new contents page which uses the currently (as @ February 2015) accepted way to vary the color and size of text. For example, read the parentheses as angle brackets,

(b style='font-size:24px')(barry31416 style='color:#009900')I. (i)Notions from Set Theory(/i) 1(/barry31416)(/b) ita - chapter 1

displays the roman numeral one heading below. Given the mindless transience of all things computer, it would seem that the long term continuous existence of all cultures is under threat of oblivion.

## Contents - All

Note: The number in parenthesis at the beginning of each entry indicates the part for which the item was required - and therefore included.

I. Notions from Set Theory 1 ( ita - chapter 1 )

sets, elements, and subsets 2
(1) ____ : Introduction
(1) ____ : Intuitive Description of Sets
(1) ____ : Union of Sets
(1) ____ : Intersection of Sets
(1) ____ : The Null Set
(1) ____ : Material Implication - if A then B
(1) ____ : Subsets

operations on sets 4
(1) ____ : The Compliment of a Set
(1) X 12 : The Compliment of a Union
(1) E 06 : The Compliment of an Intersection
(1) ____ : The Difference of two Sets
(1) ____ : Ordered Pairs
(1) ____ : The Cartesian Product of Two Sets

functions 8
(1) ____ : Function from X to Y
(1) ____ : The General Definition of Function
(1) ____ : The Restriction of a Function
(2) ____ : Composition of Functions
(1) ____ : Inverse Functions
(1) ____ : Image of a Set Under a Function
(1) ____ : Inverse Image of a Set Under a Function

finite and infinite sets 10
(1) ____ : The Natural Numbers
(1) ____ : Finite and Infinite Sets
(1) X 12 : The Image of a Union is the Union of Images
(1) X 12 : The Inverse Image of a Union
(1) X 12 : The Inverse Image of an Image
(1) X 12 : The Image of an Inverse Image

II. The Real Number System 15 ( ita - chapter 2 )

(1) ____ : Introduction
(1) ____ : The Real Number System
(1) ____ : The Field Properties
(1) ____ : The Order Property
(1) ____ : The Least Upper Bound Property
(1) ____ : Main Consequences of the Field Properties
(1) ____ : A Field of Two Elements
(1) ____ : Main Consequences of the Order Property
(1) ____ : The Field Becomes Infinite
(1) ____ : Integral Exponents
(1) ____ : Absolute Values
(1) ____ : Main Consequences of the Least Upper Bound Property
(1) ____ : Summary Page for Upper and Lower Bounds
(1) ____ : Existence of Square Roots
(1) ____ : Bigger Numbers have Bigger Square Roots

III. Metric Spaces 33 ( ita - chapter 3 )

definition of a metric space 34
(1) D 34 : Definition of a Metric Space
(1) E 34 : The Real Numbers as a Metric Space
(1) E 34 : Euclidean n-Space
(1) D 36 : Subspaces in a Metric Space
(1) P 37 : The Difference of Two Sides of a Triangle is Less than the Third Side

open and closed sets 37
(1) D 37 : Open and Closed Balls in a Metric Space
(1) D 38 : Intervals of Real Numbers
(1) D 39 : Open Sets in a Metric Space
(1) P 40 : An Open Ball is an Open Set
(1) D 40 : Closed Sets in a Metric Space
(1) P 41 : A Closed Ball is a Closed Set
(1) ____ : Examples with Real Numbers
(1) P 39 : Combinations of Open Sets
(1) P 41 : Combinations of Closed Sets
(1) E 42 : Half Spaces in Euclidean n-Space
(1) E 43 : Intervals in Euclidean n-Space
(1) D 43 : Bounded Sets
(1) P 44 : Extrema for Nonempty Bounded Closed Sets of Real Numbers
(1) X 61 : Bounded Open Sets of Real Numbers

convergent sequences 44
(1) D 45 : Convergence of a Sequence
(1) E 46 : The Constant Sequence
(1) P 46 : The Limit of a Sequence is Unique
(1) D 46 : Subsequences
(1) P 46 : Convergence of Subsequences
(1) D 47 : Bounded Sequences
(1) E 47 : Convergent Sequences are Bounded
(1) T 47 : Convergent Sequences and Closed Sets
(1) P 48 : Limit of a Product Sequence
(1) P 48 : Limits of Sum, Difference, and Quotient Sequences
(1) D 50 : Monotonic Sequences
(1) P 50 : Bounded Monotonic Sequences are Convergent

completeness 51
(1) D 51 : Cauchy Sequences
(1) P 51 : Convergent Sequences are Cauchy
(1) P 52 : Cauchy Sequences are Bounded
(1) P 52 : Convergence of a Cauchy Sequence with a Convergent Subsequence
(1) D 52 : Completeness
(1) P 52 : The Completeness of a Closed Subset of a Complete Metric Space
(1) T 53 : As a Metric Space, the Real Numbers are Complete

compactness 54
(1) D 54 : Compactness
(1) P 54 : A Compact Subset of a Metric Space is Bounded
(1) D 55 : Cluster Points
(1) ____ : Cluster Points in Open Sets of Real Numbers
(1) T 56 : Existence of a Cluster Point in Infinite Subsets of a Compact Metric Space
(1) C 56 : Convergence of a Subsequence in a Compact Metric Space
(1) C 56 : A Compact Metric Space is Complete
(1) C 56 : A Compact Subset of a Metric Space is Closed
(1) X 64 : [a,b] is Compact
(1) X 65 : More on Compactness

connectedness 59
(1) D 59 : Connectedness
(1) P 60 : A Criterion for a Set of Real Numbers to be Not Connected
(1) T 60 : Intervals of Real Numbers are Connected

IV. Continuous Functions 67 ( ita - chapter 4 )

definition of continuity 68
(1) D 68 : Continuity at a Point
(1) D 68 : Continuity on a Set
(1) ____ : Example: f(x,y) = y
(1) E 69 : A Constant Function
(2) P 70 : The Identity Function is Continuous
(1) E 70 : The Restriction of a Continuous Function is Continuous
(1) P 70 : Inverse Images of Open Sets Under Continuous Functions are Open
(2) P 71 : A Continuous Function of a Continuous Function is Continuous

continuity and limits 72
(1) E & D 72 : Limit of a Function at a Point
(1) E 74 : The Connection Between Continuity and Limits
(1) P 74 : Continuous Functions and Convergent Sequences

continuity of rational operations 75
(1) P 75 : Continuity of Rational Operations
(1) >C 76 : Limits in Rational Operations

continuous functions on a compact metric space 78
(1) T 78 : The Continuous Image of a Compact Metric Space is Compact
(1) C 78 : Max & Min - Continuous Real-valued Functions on Compact Metric Spaces
(1) D 80 : Uniform Continuity
(1) E 80 : Uniform Continuity Implies Continuity
(1) T 80 : Continuous Functions on Compact Metric Spaces are Uniformly Continuous

continuous functions on a connected metric space 82
(1) T 82 : The Continuous Image of a Connected Set is Connected
(1) C 82 : The Intermediate Value Theorem

sequences of functions 83
(1) D 83 : Convervent Sequences of Functions
(1) D 85 : Uniformly Convervent Sequences of Functions
(1) P 86 : Cauchy Criterion for Sequences of Functions
(1) T 87 : Continuity of the Limit of a Uniformly Convergent Sequence of Continuous Functions
(1) L 87 : Continuity of f,g: E to E' Implies Continuity of d'( f(p), g(p) )
(1) T 90 : A Complete Metric Space of Continuous Functions

V. Differentiation 97 ( ita - chapter 5 )

definition of derivative 98
(1) D 098 : Differentiability at a Point
(1) P 099 : Differentiability Implies Continuity
(1) D 100 : Differentiability on a Set
(1) ____: The Restriction of a Differentiable Function is Differentiable

rules of differentiation 100
(1) E 100 : The Derivative of a Constant Function is Zero
(1) E 100 : The Derivative of the Identity Function is One
(1) P 101 : Rules of Differentiation
(1) ____: The Derivative of a Product of n Functions
(2) P 103 : The Chain Rule

the mean value theorem 103
(1) P 103 : The Derivative at Extrema is Zero
(1) L 104 : Rolle's Theorem
(1) T 105 : The Mean Value Theorem
(2) D 105 : Increasing and Decreasing Functions
(2) C 105 : Increasing and Decreasing Functions - The Derivative's Implication

VI. Riemann Integration 111 ( ita - chapter 6 )

definitions and examples 112
(1) D 112 : The Riemann Sum
(1) D 112 : The Riemann Integral
(1) E 114 : Integral of a Constant Function
(1) E 114 : Integral of a Function with Only One Non-zero Point
(1) E 114 : Integral of a Unit Step

linearity and order properties of the integral 116
(1) P 116 : Integrals of Sums and Constant Multiples of Functions
(1) P 117 : The Integral of a Non-negative Function is Non-negative
(1) C 117 : Integration of an Inequality
(1) C 118 : Bounds for an Integral

existence of the integral 118
(1) L 118 : Integrability
(1) D 119 : Step Functions
(1) L 119 : Step Functions are Integrable
(1) P 120 : General Condition for Riemann Integrability
(1) T 123 : Continuous Functions are Riemann Integrable

the fundamental theorem of calculus 123
(1) P 123 : Subintervals in Riemann Integration
(1) D 125 : Negation of a Riemann Integral
(1) C 125 : Integration from a to b to c - Regardless of Order
(1) T 126 : The Fundamental Theorem of Calculus
(1) C 127 : The Integral in Terms of its Antiderivative

the logarithmic and exponential functions 128
(2) D 128 : Definition of the Logarithm
(2) P 128 : Properties of the Logarithm Function
(2) P 129 : Definition of the Exponential Function

VII. Interchange of Limit Operations 137 ( ita - chapter 7 )

differentiation under the integral sign 159
(1) D 159 : The Partial Derivative

VIII. The Method of Successive Approximations 169 ( ita - chapter 8 )

the fixed point theorem 170
(1) T 170 : The Fixed Point Theorem

uniqueness of the real number system 29
(1) ____: b-mal Expansions of Real Numbers
(1) ____: Every Infinite b-mal Represents a Real Number
(1) ____: Periodic Infinite b-mals
(1) Discussion 29 : Uniqueness of the Real Number System

the simplest case of the implicit function theorem 173
(1) T 174 : The Simplest Case of The Implicit Function Theorem
(1) C 176 : The Inverse Function Theorem

existence and uniqueness theorems for ordinary differential equations 177
(1) T 178 : dy/dx = f(x,y)

IX. Partial Differentiation 193 ( ita - chapter 9 )

X. Multiple Integrals 215 ( ita - chapter 10 )