# Math 424 - Fundamental Concepts of Analysis I

See UW General Catalog for course description and prerequisite information.

#### Textbooks

• Advanced Calculus (Second Edition) by Patrick M. Fitzpatrick
• Principles of Mathematical Analysis (Third Edition) by Walter Rudin

### Suggested Syllabus

#### Continuity

• epsilon-delta definition and sequence definition of continuity
• epsilon-delta and sequence definitions of a limit
• pointwise convergence, uniform convergence, uniformly Cauchy sequences
• uniform limits and continuity

#### Differentiation

• definition of the derivative
• tangent line approximation
• algebra of derivatives and the chain rule
• Rolle’s Theorem and the Mean Value Theorem
• higher derivatives and Taylor’s Theorem
• Intermediate Value Theorem for derivatives

#### The Riemann-Stieltjes Integral

• definition of the Riemann-Stieltjes integral
• integrability of continuous and monotone functions
• properties of the integral
• change of variables
• Fundamental Theorem of Calculus
• integration by parts
• Mean Value Theorem for Riemann integrals

#### Uniform Convergence and Power Series

• uniform convergence and integration
• uniform convergence and differentiation
• Weierstrass M-test
• power series and radius of convergence
• differentiation of power series
• Weierstrass Approximation Theorem