# MAT 401 Advanced Analysis I

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### Course Description

Elements of set theory, numerical sequences and series, continuity and differentiability of functions of one and several variables.

MAT 401 meets for three hours of lecture per week.

### Prerequisites

MAT 211 and MAT 271 with grade C or better.

### Objectives

After completing MAT 401 the student should be able to

• know, understand and use techniques and concepts from elementary topology and real analysis: open, closed, and compact sets, least-upper-bounds, Euclidean norm, triangle and Schwartz inequalities, convergence, limits, Cauchy sequences, epsilon-delta proof techniques, uniform convergence, continuity, differentiability.
• prove basic results about functions on R^n and sets and sequences in R^n. These basic results include the following
• a sequence converges in R^n if and only if it is Cauchy
• bounded sequences have Cauchy subsequences
• closed bounded sets in R^n are compact
• sums, products, quotients of convergent sequences are convergent, sums, products, quotients, compositions of continuous, (uniformly continuous) functions on open sets are continuous, (uniformly continuous), where defined
• on compact sets, continuous functions are uniformly continuous and have maxima and minima
• differentiable functions are continuous
• method for prooving from first principles that a given function is continuous or differentiable at a given point
• the student must be able to solve concrete problems and write clear and coherent mathematical arguments incorporating these techniques and concepts.

### Expected outcomes

Students should be able to demonstrate through written assignments, tests, and/or oral presentations, that they have achieved the objectives of MAT 401.

### Method of Evaluating Outcomes

Evaluations are based on homework, class participation, short tests and scheduled examinations covering students' understanding of elementary topology, real analysis, and related topics that are covered in MAT 401.

### Text

Advanced Calculus, by R. Creighton Buck. McGraw-Hill, 1978.

• 1. Sets and Functions
• 1.1. Introduction
• 1.2. R and R^n
• 1.3. Distance
• 1.4. Functions
• 1.5. Topological Terminology
• 1.6. Sequences
• 1.7. Consequences of the Monotonic-Sequence Property
• 1.8. Compact Sets
• 2. Continuity
• 2.1. Preview
• 2.2. Basic Definitions
• 2.3. Uniform Continuity
• 2.4. Implications of Continuity
• 2.5. Limits of Functions
• 2.6. Discontinuities
• 2.7. Inverses for Functions of One Variable
• 3. Differentiation
• 3.1. Preview.
• 3.2. Mean Value Theorems and L'Hospital's Rule
• 3.3. Derivatives for Functions on R^n

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in MAT 401. The instructor determines the relative weights of these factors.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates and Make-Up Work

Due dates and policy regarding make-up work are set by the instructor.

### Schedule of Examinations

The instructor sets all test dates except the date of the final exam. The final exam is given at the date and time announced in the Schedule of Classes.