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MATH 422 Syllabus

Real Analysis I

Revised: September, 2014

Course Description

Sequences of real numbers, continuous functions, and differentiation. Prerequisite: MATH 250 and MATH 255. Three Semester Hours.


Math250 and Math255.

Student Learning Objectives

By the end of the course students will be able to:

  • Use the definitions of convergence as they apply to sequences, series, and functions;
  • Determine the continuity, differentiability, and integrability of functions defined on subsets of the real line;
  • Produce rigorous proofs of results that arise in the context of real analysis; and
  • Write proofs of theorems that meet rigorous standards based on content, precision, and style.


Gordon. Real Analysis, A First Course, Second Edition. Addison-Wesley, 2002.

Grading Procedure

Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.

Attendance Policy

Attendance policy is left to the discretion of individual instructors, subject to general university policy.

Course Outline

  • Chapter 1: Real Numbers (6 days)
Completeness; countable and uncountable sets; real valued functions
  • Chapter 2: Sequences (10 days)
Convergent monotone and Cauchy sequences; subsequences; Bolzano-Weierstrass
  • Chapter 3: Limits and Continuity (14 days)
Limit theorems; one-sided and infinite limits; continuous functions; intermediate and extreme values; uniform continuity; monotone functions
  • Chapter 4: Differentiation (10 days)
The definition and rules of differentiation; mean value and L'Hopital
  • Chapter 5: Integration (5 days, if time allows)
Riemann Integral; conditions for Riemann integrability

Times above include review and testing.

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