Basic Analysis: Functions of a Real Variable

Cover
Half Title
Title Page
Copyright Page
Dedication
Abstract
Acknowledgments
Table of Contents
I: Introduction
	1: Introduction
		1.1 Table of Contents
II: Understanding Smoothness
	2: Proving Propositions
		2.1 Mathematical Induction
			2.1.1 Homework
		2.2 More Examples
		2.3 More Abstract Proofs and Even Trickier POMIs
			2.3.1 Homework
		2.4 Some Contradiction Proofs
		2.5 Triangle Inequalities.
			2.5.1 A First Look at the Cauchy - Schwartz Inequality
			2.5.2 Homework
		2.6 The Supremum and Infimum of a Set of Real Numbers
			2.6.1 Bounded Sets
			2.6.2 Least Upper Bounds and Greatest Lower Bounds
			2.6.3 The Completeness Axiom and Consequences
			2.6.4 Some Basic Results
			2.6.5 Homework
	3: Sequences of Real Numbers
		3.1 Basic Definitions
		3.2 The Definition of a Sequence
			3.2.1 Homework
		3.3 The Convergence of a Sequence
			3.3.1 Proofs of Divergence
			3.3.2 Uniqueness of Limits and So Forth
			3.3.3 Proofs of Convergence
			3.3.4 Homework
		3.4 Sequence Spaces
			3.4.1 Limit Theorems
			3.4.2 Some Examples
			3.4.3 Homework
	4: Bolzano - Weierstrass Results
		4.1 Bounded Sequences with a Finite Range
		4.2 Sequences with an Infinite Range
			4.2.1 Extensions to ℜ2
		4.3 Bounded Infinite Sets
			4.3.1 Homework
			4.3.2 Bounded Infinite Sets Have at Least One Accumulation Point
			4.3.3 Cluster Points of a Set
		4.4 More Set and Sequence Related Topics
			4.4.1 Homework
			4.4.2 Some Set Properties
		4.5 The Notion of Sequential Compactness
			4.5.1 Sequential Compactness and the Existence of Extrema
			4.5.2 Homework
		4.6 The Deeper Structure of Sequences
			4.6.1 The Limit Inferior and Limit Superior of a Sequence
			4.6.2 Limit Inferior* Star and Limit Superior*
			4.6.3 Homework
	5: Topological Compactness
		5.1 More Set Theory
		5.2 The Notion of Topological Compactness
			5.2.1 Finite Closed Intervals and Topological Compactness
			5.2.2 Homework
			5.2.3 The Equivalence of Topological Compactness and Closed and Bounded
			5.2.4 Homework
	6: Function Limits
		6.1 The Limit Inferior and Limit Superior of Functions
			6.1.1 A Poorly Behaved Function
			6.1.2 A Function with Only One Set of Cluster Points That Is One Value
			6.1.3 A Function with All Cluster Point Sets Having Two Values
			6.1.4 Homework
			6.1.5 Examples of Cluster Points of Functions
			6.1.6 C∞ Bump Functions
		6.2 Limits of Functions
	7: Continuity
		7.1 Continuity
			7.1.1 Homework
			7.1.2 Dirichlet’s Function: Lack of Continuity at Each Rational Number
		7.2 Limit Examples
			7.2.1 Right and Left Continuity at a Point
			7.2.2 Homework
		7.3 Limit and Continuity Proofs
		7.4 The Algebra of Limits
			7.4.1 The Algebra of Continuity
		7.4.2 Homework
	8: Consequences of Continuity on Intervals
		8.1 Domains of Continuous Functions
		8.2 The Intermediate Value Theorem
		8.3 Continuous Images of Compact Sets
		8.4 Continuity in Terms of Inverse Images of Open Sets
			8.4.1 Homework
	9: Lower Semicontinuous and Convex Functions
		9.1 Lower Semicontinuous Functions
		9.2 Convex Functions
			9.2.1 Homework
		9.3 More on Convex Functions
		9.4 Subdifferentials
			9.4.1 Homework
	10: Basic Differentiability
		10.1 An Introduction to Smoothness
		10.2 A Basic Evolutionary Model
			10.2.1 A Difference Equation
			10.2.2 The Functional Form of the Frequency
			10.2.3 Abstracting Generation Time
		10.3 Sneaking Up on Differentiability
			10.3.1 Change and More Change
			10.3.2 Homework
		10.4 Limits, Continuity and Right and Left Hand Slope Limits
		10.5 Differentiability
			10.5.1 Homework
	11: The Properties of Derivatives
		11.1 Simple Derivatives
		11.2 The Product Rule
		11.3 The Quotient Rule
			11.3.1 Homework
		11.4 Function Composition and Continuity
		11.5 Chain Rule
			11.5.1 Homework
		11.6 Sin, Cos and All That!
			11.6.1 Examples
			11.6.2 A New Power Rule
			11.6.3 More Complicated Derivatives
			11.6.4 Homework
	12: Consequences of Derivatives
		12.1 Taylor Polynomials
			12.1.1 Zeroth Order Taylor Polynomials
			12.1.2 The Order One Taylor Polynomial
			12.1.3 Second Order Approximations
			12.1.4 Homework
			12.1.5 Taylor Polynomials in General
		12.2 Differences of Functions
			12.2.1 Two Close Cosines
			12.2.2 Two Close Exponentials
			12.2.3 Approximations in a Cancer Model
			12.2.4 Homework
	13: Exponential and Logarithm Functions
		13.1 The Number e: First Definition
			13.1.1 Homework
		13.2 The Number e: Second Definition
		13.3 The Exponential Function
			13.3.1 The Properties of the Exponential Function
		13.4 Taylor Series
			13.4.1 Homework
		13.5 Inverse Functions
			13.5.1 Homework
			13.5.2 Continuity of the Inverse Function
		13.6 L’Hôpital’s Rules
	14: Extremal Theory for One Variable
		14.1 Extremal Values
		14.2 The First Derivative Test
		14.3 Cooling Models
			14.3.1 Homework
	15: Differentiation in ℜ2 and ℜ3
		15.1 ℜ2 and ℜ3
		15.2 Functions of Two Variables
			15.2.1 Homework
		15.3 Continuity
			15.3.1 Homework
		15.4 Partial Derivatives
		15.5 Tangent Planes
			15.5.1 Homework
		15.6 Derivatives in 2D!
		15.7 Chain Rule
			15.7.1 Homework
		15.8 Tangent Plane Approximation Error
		15.9 Hessian Approximations
			15.9.1 Homework
		15.10 Partials Existing Does Not Necessarily Imply Differentiable
		15.11 When Do Mixed Partials Match?
			15.11.1 Homework
	16: Multivariable Extremal Theory
		16.1 Extrema Ideas
			16.1.1 Saddle Points
			16.1.2 Homework
		16.2 Symmetric Problems
			16.2.1 A Canonical Form for a Symmetric Matrix
			16.2.2 Signed Definite Matrices
		16.3 A Deeper Look at Extremals
			16.3.1 Extending to Three Variables
			16.3.2 Homework
III: Integration and Sequences of Functions
	17: Uniform Continuity
		17.1 Uniform Continuity
			17.1.1 Homework
		17.2 Uniform Continuity and Differentiation
			17.2.1 Homework
	18: Cauchy Sequences of Real Numbers
		18.1 Cauchy Sequences
		18.2 Completeness
		18.3 The Completeness of the Real Numbers
		18.4 Uniform Continuity and Compact Domains
			18.4.1 Homework
	19: Series of Real Numbers
		19.1 Series of Real Numbers
		19.2 Basic Facts about Series
			19.2.1 An ODE Example: Things to Come
			19.2.2 Homework
		19.3 Series with Non-Negative Terms
		19.4 Comparison Tests
		19.5 p Series
		19.6 Examples
			19.6.1 Homework
	20: Series in General
		20.1 Some Specific Tests for Convergence
			20.1.1 Geometric Series
			20.1.2 The Ratio Test
			20.1.3 The Root Test
			20.1.4 Examples
			20.1.5 A Taste of Power Series
		20.2 Cauchy - Schwartz and Minkowski Inequalities
			20.2.1 Conjugate Exponents
			20.2.2 Hölder’s Inequality
			20.2.3 Minkowski’s Inequality
			20.2.4 The ℓp Spaces and their Metric
			20.2.5 Homework
		20.3 Convergence in a Metric Space
		20.4 The Completeness of Some Spaces
			20.4.1 The Completeness of (C([a, b]), ∥⋅∥∞)
			20.4.2 The Completeness of (ℓ∞; ∥⋅∥∞)
			20.4.3 The Completeness of (ℓp, ∥⋅∥p)
			20.4.4 Homework
	21: Integration Theory
		21.1 Basics
			21.1.1 Partitions of [a, b]
			21.1.2 Riemann Sums
			21.1.3 Riemann Sums in Octave
			21.1.4 Graphing Riemann Sums
			21.1.5 Uniform Partitions and Riemann Sums
			21.1.6 Refinements of Partitions
			21.1.7 Homework
			21.1.8 Uniform Partition Riemann Sums and Convergence
		21.2 Defining The Riemann Integral
			21.2.1 Fundamental Integral Estimates
	22: Existence of the Riemann Integral and Properties
		22.1 The Darboux Integral
			22.1.1 Upper and Lower Darboux Sums
			22.1.2 Upper and Lower Darboux Integrals and Darboux Integrability
		22.2 The Darboux and Riemann Integrals are Equivalent
		22.3 Properties
		22.4 Riemann Integrable Functions
			22.4.1 Homework
		22.5 More Properties
	23: The Fundamental Theorem of Calculus (FTOC)
		23.1 The Proof of the FTOC
		23.2 The Cauchy FTOC
		23.3 Integral Mean Value Results
		23.4 Approximation of the Riemann Integral
			23.4.1 Homework
		23.5 Substitution
		23.6 The Natural Logarithm Function
			23.6.1 Logarithm Functions
			23.6.2 Worked Out Examples: Integrals
			23.6.3 The Exponential Function
			23.6.4 Positive Powers of e
			23.6.5 Negative Integer Powers of e
			23.6.6 Adding Natural Logarithms
			23.6.7 Logarithm Properties
			23.6.8 The Exponential Function Properties
		23.7 When Do Riemann Integrals Match?
			23.7.1 When Do Two Functions Have the Same Integral?
			23.7.2 Calculating the FTOC Function
	24: Convergence of Sequences of Functions
		24.1 The Weierstrass Approximation Theorem
			24.1.1 Bernstein Code Implementation
			24.1.2 A Theoretical Example
		24.2 The Convergence of a Sequence of Functions
			24.2.1 Homework
			24.2.2 More Complicated Examples
			24.2.3 What Do Our Examples Show Us?
			24.2.4 Sequence Convergence and Integration
			24.2.5 More Conclusions
		24.3 Basic Theorems
			24.3.1 Limit Interchange Theorems for Integration
			24.3.2 Homework
		24.4 The Interchange Theorem for Differentiation
	25: Series of Functions and Power Series
		25.1 The Sum of Powers of t Series
			25.1.1 More General Series
			25.1.2 Homework
		25.2 Uniform Convergence Tests
			25.2.1 Homework
		25.3 Integrated Series
			25.3.1 Homework
		25.4 Differentiating Series
			25.4.1 Higher Derived Series
			25.4.2 Homework
		25.5 Power Series
			25.5.1 Homework
	26: Riemann Integration: Discontinuities and Compositions
		26.1 Compositions of Riemann Integrable Functions
		26.2 Content Zero
			26.2.1 The Riemann - Lebesgue Lemma
			26.2.2 Equivalence Classes of Riemann Integrable Functions
	27: Fourier Series
		27.1 Vector Space Notions
			27.1.1 Homework
			27.1.2 Inner Products
			27.1.3 Homework
		27.2 Fourier Series
		27.3 Fourier Series Components
			27.3.1 Orthogonal Functions
			27.3.2 Fourier Coefficients Revisited
		27.4 Fourier Series Convergence
			27.4.1 Rewriting SN(x)
			27.4.2 Rewriting SN – f
			27.4.3 Convergence
		27.5 Fourier Sine Series
			27.5.1 Homework
		27.6 Fourier Cosine Series
			27.6.1 Homework
		27.7 More Convergence Fourier Series
			27.7.1 Bounded Coefficients
			27.7.2 The Derivative Series
			27.7.3 Derivative Bounds
			27.7.4 Uniform Convergence for f
			27.7.5 Extending to Fourier Sine and Fourier Cosine Series
		27.8 Code Implementation
			27.8.1 Inner Products Revisited
			27.8.2 General GSO
			27.8.3 Fourier Approximations
			27.8.4 Testing the Approximations
	28: Applications
		28.1 Solving Differential Equations
			28.1.1 Homework
		28.2 Some ODE Techniques
			28.2.1 Variation of Parameters
			28.2.2 Homework
			28.2.3 Boundary Value Problems
		28.3 Linear Partial Differential Equations
			28.3.1 Determining the Separation Constant
			28.3.2 Convergence Analysis for Fourier Series Revisited
			28.3.3 Fourier Series Convergence Analysis
			28.3.4 Homework
			28.3.5 Convergence of the Cable Solution: Simplified Analysis
			28.3.6 Homework
		28.4 Power Series for ODE
			28.4.1 Constant Coefficient ODEs
			28.4.2 Homework
			28.4.3 Polynomial Coefficient ODEs
			28.4.4 Homework
IV: Summing It All Up
	29: Summary
V: References
VI: Detailed Index
	Index