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.

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

** I. ** ( 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) **:** The Compliment of a Union

(1) **:** 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) **:** The Image of a Union is the Union of Images

(1) **:** The Inverse Image of a Union

(1) **:** The Inverse Image of an Image

(1) **:** The Image of an Inverse Image

** II. ** ( 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. ** ( ita - chapter 3 )

*definition of a metric space* 34

(1) **:** Definition of a Metric Space

(1) **:** The Real Numbers as a Metric Space

(1) **:** Euclidean *n*-Space

(1) **:** Subspaces in a Metric Space

(1) **:** The Difference of Two Sides of a Triangle is Less than the Third Side

*open and closed sets* 37

(1) **:** Open and Closed Balls in a Metric Space

(1) **:** Intervals of Real Numbers

(1) **:** Open Sets in a Metric Space

(1) **:** An Open Ball is an Open Set

(1) **:** Closed Sets in a Metric Space

(1) **:** A Closed Ball is a Closed Set

(1) **____** **:** Examples with Real Numbers

(1) **:** Combinations of Open Sets

(1) **:** Combinations of Closed Sets

(1) **:** Half Spaces in Euclidean *n*-Space

(1) **:** Intervals in Euclidean *n*-Space

(1) **:** Bounded Sets

(1) **:** Extrema for Nonempty Bounded Closed Sets of Real Numbers

(1) **:** Bounded Open Sets of Real Numbers

*convergent sequences* 44

(1) **:** Convergence of a Sequence

(1) **:** The Constant Sequence

(1) **:** The Limit of a Sequence is Unique

(1) **:** Subsequences

(1) **:** Convergence of Subsequences

(1) **:** Bounded Sequences

(1) **:** Convergent Sequences are Bounded

(1) **:** Convergent Sequences and Closed Sets

(1) **:** Limit of a Product Sequence

(1) **:** Limits of Sum, Difference, and Quotient Sequences

(1) **:** Monotonic Sequences

(1) **:** Bounded Monotonic Sequences are Convergent

*completeness* 51

(1) **:** Cauchy Sequences

(1) **:** Convergent Sequences are Cauchy

(1) **:** Cauchy Sequences are Bounded

(1) **:** Convergence of a Cauchy Sequence with a Convergent Subsequence

(1) **:** Completeness

(1) **:** The Completeness of a Closed Subset of a Complete Metric Space

(1) **:** As a Metric Space, the Real Numbers are Complete

*compactness* 54

(1) **:** Compactness

(1) **:** A Compact Subset of a Metric Space is Bounded

(1) **:** Cluster Points

(1) **____** **:** Cluster Points in Open Sets of Real Numbers

(1) **:** Existence of a Cluster Point in Infinite Subsets of a Compact Metric Space

(1) **:** Convergence of a Subsequence in a Compact Metric Space

(1) **:** A Compact Metric Space is Complete

(1) **:** A Compact Subset of a Metric Space is Closed

(1) **:** [*a*,*b*] is Compact

(1) **:** More on Compactness

*connectedness* 59

(1) **:** Connectedness

(1) **:** A Criterion for a Set of Real Numbers to be Not Connected

(1) **:** Intervals of Real Numbers are Connected

** IV. ** ( ita - chapter 4 )

*definition of continuity* 68

(1) **:** Continuity at a Point

(1) **:** Continuity on a Set

(1) **____** **:** Example: *f*(*x*,*y*) = *y*

(1) **:** A Constant Function

(2) **:** The Identity Function is Continuous

(1) **:** The Restriction of a Continuous Function is Continuous

(1) **:** Inverse Images of Open Sets Under Continuous Functions are Open

(2) **:** A Continuous Function of a Continuous Function is Continuous

*continuity and limits* 72

(1) **:** Limit of a Function at a Point

(1) **:** The Connection Between Continuity and Limits

(1) **:** Continuous Functions and Convergent Sequences

*continuity of rational operations* 75

(1) **:** Continuity of Rational Operations

(1) **:** Limits in Rational Operations

*continuous functions on a compact metric space* 78

(1) **:** The Continuous Image of a Compact Metric Space is Compact

(1) **:** Max & Min - Continuous Real-valued Functions on Compact Metric Spaces

(1) **:** Uniform Continuity

(1) **:** Uniform Continuity Implies Continuity

(1) **:** Continuous Functions on Compact Metric Spaces are Uniformly Continuous

*continuous functions on a connected metric space* 82

(1) **:** The Continuous Image of a Connected Set is Connected

(1) **:** The Intermediate Value Theorem

*sequences of functions* 83

(1) **:** Convervent Sequences of Functions

(1) **:** Uniformly Convervent Sequences of Functions

(1) **:** Cauchy Criterion for Sequences of Functions

(1) **:** Continuity of the Limit of a Uniformly Convergent Sequence of Continuous Functions

(1) **:** Continuity of *f*,*g***:** *E* to *E'* Implies Continuity of d'( *f*(*p*), *g*(*p*) )

(1) **:** A Complete Metric Space of Continuous Functions

** V. ** ( ita - chapter 5 )

*definition of derivative* 98

(1) **:** Differentiability at a Point

(1) **:** Differentiability Implies Continuity

(1) **:** Differentiability on a Set

(1) **____****:** The Restriction of a Differentiable Function is Differentiable

*rules of differentiation* 100

(1) **:** The Derivative of a Constant Function is Zero

(1) **:** The Derivative of the Identity Function is One

(1) **:** Rules of Differentiation

(1) **____****:** The Derivative of a Product of *n* Functions

(2) **:** The Chain Rule

*the mean value theorem* 103

(1) **:** The Derivative at Extrema is Zero

(1) **:** Rolle's Theorem

(1) **:** The Mean Value Theorem

(2) **:** Increasing and Decreasing Functions

(2) **:** Increasing and Decreasing Functions - The Derivative's Implication

** VI. ** ( ita - chapter 6 )

*definitions and examples* 112

(1) **:** The Riemann Sum

(1) **:** The Riemann Integral

(1) **:** Integral of a Constant Function

(1) **:** Integral of a Function with Only One Non-zero Point

(1) **:** Integral of a Unit Step

*linearity and order properties of the integral* 116

(1) **:** Integrals of Sums and Constant Multiples of Functions

(1) **:** The Integral of a Non-negative Function is Non-negative

(1) **:** Integration of an Inequality

(1) **:** Bounds for an Integral

*existence of the integral* 118

(1) **:** Integrability

(1) **:** Step Functions

(1) **:** Step Functions are Integrable

(1) **:** General Condition for Riemann Integrability

(1) **:** Continuous Functions are Riemann Integrable

*the fundamental theorem of calculus* 123

(1) **:** Subintervals in Riemann Integration

(1) **:** Negation of a Riemann Integral

(1) **:** Integration from *a* to *b* to *c* - Regardless of Order

(1) **:** The Fundamental Theorem of Calculus

(1) **:** The Integral in Terms of its Antiderivative

*the logarithmic and exponential functions* 128

(2) **:** Definition of the Logarithm

(2) **:** Properties of the Logarithm Function

(2) **:** Definition of the Exponential Function

*differentiation under the integral sign* 159

(1) **:** The Partial Derivative

** VIII. ** ( ita - chapter 8 )

*the fixed point theorem* 170

(1) **:** 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) **:** Uniqueness of the Real Number System

*the simplest case of the implicit function theorem* 173

(1) **:** The Simplest Case of The Implicit Function Theorem

(1) **:** The Inverse Function Theorem

*existence and uniqueness theorems for ordinary differential equations* 177

(1) **:** d*y*/dx = f(*x*,*y*)

** IX. ** ( ita - chapter 9 )

** X. ** ( ita - chapter 10 )