Why Does Lang Define Limits Nonstandardly in Undergraduate Analysis?

[micromass]

• Author: Serge Lang
• Title: Undergraduate Analysis
• Amazon Link: https://www.amazon.com/dp/1441928537/?tag=pfamazon01-20
• Prerequisities: Calculus, Proofs
• Level: Undergrad

Table of Contents:

[LIST]
[*] Foreword
[*] Review of Calculus
[LIST]
[*] Sets and Mappings
[LIST]
[*] Sets
[*] Mappings
[*] Natural Numbers and Induction
[*] Denumerable Sets
[*] Equivalence Relations
[/LIST]
[*] Real Numbers
[LIST]
[*] Algebraic Axioms
[*] Ordering Axioms
[*] Integers and Rational Numbers
[*] The Completeness Axiom
[/LIST]
[*] Limits and Continuous Functions
[LIST]
[*] Sequences of Numbers
[*] Functions and Limits
[*] Limits with Infinity
[*] Continuous Functions
[/LIST]
[*] Differentiation
[LIST]
[*] Properties of the Derivative
[*] Mean Value Theorem
[*] Inverse Functions
[/LIST]
[*] Elementary Functions
[LIST]
[*] Exponential
[*] Logarithm
[*] Sine and Cosine
[*] Complex Numbers
[/LIST]
[*] The Elementary Real Integral
[LIST]
[*] Characterization of the Integral
[*] Properties of the Integral
[*] Taylor's Formula
[*] Asymptotic Estimates and Stirling's Formula
[/LIST]
[/LIST]
[*] Convergence
[LIST]
[*] Normed Vector Spaces
[LIST]
[*] Vector Spaces
[*] Normed Vector Spaces
[*] n-Space and Function Spaces
[*] Completeness
[*] Open and Closed Sets
[/LIST]
[*] Limits
[LIST]
[*] Basic Properties
[*] Continuous Maps
[*] Limits in Function Spaces
[*] Completion of a Normed Vector Space
[/LIST]
[*] Compactness
[LIST]
[*] Basic Properties of Compact Sets
[*] Continuous Maps on Compact Sets
[*] Algebraic Closure of the Complex Numbers
[*] Relation with Open Coverings
[/LIST]
[*] Series
[LIST]
[*] Basic Definitions
[*] Series of Positive Numbers
[*] Non-Absolute Convergence
[*] Absolute Convergence in Vector Spaces
[*] Absolute and Uniform Convergence
[*] Power Series
[*] Differentiation and Integration of Series
[/LIST]
[*] The Integral in One Variable
[LIST]
[*] Extension Theorem for Linear Maps
[*] Integral of Step Maps
[*] Approximation by Step Maps
[*] Properties of the Integral
[*] Appendix: The Lebesgue Integral
[*] The Derivative
[*] Relation Between the Integral and the Derivative
[*] Interchanging Derivatives and Integrals
[/LIST]
[/LIST]
[*] Applications of the Integral
[LIST]
[*] Approximation with Convolutions
[LIST]
[*] Dirac Sequences
[*] The Weierstrass Theorem
[/LIST]
[*] Fourier Series
[LIST]
[*] Hermitian Products and Orthogonality
[*] Trigonometric Polynomials as a Total Family
[*] Explicit Uniform Approximation
[*] Pointwise Convergence
[/LIST]
[*] Improper Integrals
[LIST]
[*] Definition
[*] Criteria for Convergence
[*] Interchanging Derivatives and Integrals
[*] The Heat Kernel
[/LIST]
[*] The Fourier Integral
[LIST]
[*] The Schwartz Space
[*] The Fourier Inversion Formula
[*] An Example of Fourier Transform not in the Schwartz Space
[/LIST]
[/LIST]
[*] Calculus in Vector Spaces
[LIST]
[*] Functions on n-Space
[LIST]
[*] Partial Derivatives
[*] Differentiability and the Chain Rule
[*] Potential Functions
[*] Curve Integrals
[*] Taylor's Formula
[*] Maxima and the Derivative
[/LIST]
[*] The Winding Number and Global Potential Functions
[LIST]
[*] Another Description of the Integral Along a Path
[*] The Winding Number and Homology
[*] Proof of the Global Integrability Theorem
[*] The Integral Over Continuous Paths
[*] The Homotopy Form of the Integrability Theorem
[*] More on Homotopies
[/LIST]
[*] Derivatives in Vector Spaces
[LIST]
[*] The Space of Continuous Linear Maps
[*] The Derivative as a Linear Map
[*] Properties of the Derivative
[*] Mean Value Theorem
[*] The Second Derivative
[*] Higher Derivatives and Taylor's Formula
[*] Partial Derivatives
[*] Differentiating Under the Integral Sign
[/LIST]
[*] Inverse Mapping Theorem
[LIST]
[*] The Shrinking Lemma
[*] Inverse Mappings, Linear Case
[*] The Inverse Mapping Theorem 
[*] Implicit Functions and Charts
[*] Product Decompositions
[/LIST]
[*] Ordinary Differential Equations
[LIST]
[*]  Local Existence and Uniqueness
[*] Approximate Solutions
[*] Linear Differential Equations
[*] Dependence on Initial Conditions
[/LIST]
[/LIST]
[*] Multiple Integration
[LIST]
[*] Multiple Integrals
[LIST]
[*] Elementary Multiple Integration
[*] Criteria for Admissibility
[*] Repeated Integrals
[*] Change of Variables
[*] Vector Fields on Spheres
[/LIST]
[*] Differential Forms
[LIST]
[*] Definitions
[*] Stokes' Theorem for a Rectangle
[*] Inverse Image of a Form
[*] Stokes' Formula for Simplices
[/LIST]
[/LIST]
[*] Appendix
[*] Index
[/LIST]

 

[mathwonk]

I have had this book since 1970, when it was titled Analysis I. Like all of Lang's books the explanations are succinct and clear, there are only a few but helpful exercises, and everything is proved. One thing I recall especially from this book is how simple, actually trivial, he made the definition of the Riemann integral look.

I.e. you want the integral to be monotone: bigger functions should have bigger integrals. And you want constant functions (rectangles) to have integral equal to "base times height".

You also want the integral to add when you break up into smaller intervals.

You are done. There is no other way to define it except as the unique number caught between the areas of all rectangles lying over and under the graph. (Maybe some people call this the Darboux integral.)

On a good day, this is what Lang does for you, you scratch your head and say "why did i think this was so hard or so complicated?"

But you don't get a lot of computational practice.

 

[jbunniii]

I like this book, but I find it frustrating when Lang makes eccentric nonstandard definitions, such as defining the limit of a function at a point c such that if f(c) is defined, the limit must equal f(c). Therefore, for Lang, if
$$f(x) = \begin{cases} 0 & \textrm{ if } x \neq 0 \\
1 & \textrm { if } x = 0\end{cases}$$
then $\lim_{x \rightarrow 0}f(x)$ does not exist. Whereas if we remove $x=0$ from the domain of the function, then the limit exists and equals zero. (See page 44 in the 2nd edition if you don't believe me.)