# Math 341: Theory of Analysis 1

### Title

Theory of Analysis 1.

(3:3:0)

F, W

### Description

Rigorous treatment of calculus of a single real variable: topology, order, completeness of real numbers; continuity, differentiability, integrability, and convergence of functions.

## Desired Learning Outcomes

The main purpose of this course is to provide students with an understanding of the real number system and of real-valued functions of a single real variable, with the focus being on the theoretical and logical foundations of single-variable calculus. A secondary purpose of this course is to reinforce students' prior training in discovering and writing valid mathematical proofs.

### Prerequisites

The prerequisites for this course are Math 113 and 290. The first is to ensure that the student has had a complete course in single-variable calculus at the introductory level, and the second is to ensure that the student knows how to read and write proofs and is familiar with the fundamental objects of advanced mathematics.

### Minimal learning outcomes

Outlined below are topics that all successful Math 341 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.

1. Basic properties of R
• Characterization as a complete, ordered field
• Archimedean Property
• Density of Q and R \ Q
• Uncountability of each interval
2. Convergence of real sequences and series
• Convergence of bounded monotonic sequences
• Algebraic and order rules for limits
• Cauchy criterion for sequences and series
• Common convergence tests for series
• Convergence of rearranged series
3. Basic topology of R
• Open and closed sets
• Limit points and limits
• Characterizations of compactness
• Sequential
• Open coverings
• The Heine-Borel Theorem
• The Bolzano-Weierstrass Theorem
• Connectedness
4. Continuity of f: DRR
• Functional limits
• Metric, sequential, and topological characterizations of continuity
• Combinations of continuous functions
• Continuity vs. uniform continuity
• Preservation of compactness
• The Extreme Value Theorem
• Uniform continuity for compact domains
• Preservation of connectedness
• The Intermediate Value Theorem
5. Differentiability of f: DRR
• Algebraic differential rules
• Chain rule
• Characterizing extrema
• Rolle's Theorem
• The Mean Value Theorem
• The Generalized Mean Value Theorem
• L'Hôpital's Rule
6. Integrability of f: [a,b] → R
• The Darboux integral
• The Riemann integral
• Integrability of continuous functions
• Integrability of monotonic functions
• Rules for combining and comparing integrals
• The Fundamental Theorem(s) of Calculus
7. Convergence of sequences and series of functions
• Pointwise vs. uniform convergence
• Relation of uniform convergence to:
• Continuity
• Differentiation
• Integration
• The Weierstrass M-Test
• The Weierstrass Approximation Theorem
• Power series
• Continuity
• Absolute and uniform convergence
• Termwise differentiability
• Taylor's Theorem
• Analyticity vs. smoothness

### Textbooks

Possible textbooks for this course include (but are not limited to):

• Stephen Abbott, Understanding Analysis, Springer, 2001.