MATH 412 Introduction to Real Analysis (3)

Catalog Description:

A rigorous development of the real number system from its foundational axioms through calculus including limits, continuity, differentiation, and integration. (Offered spring semester only.)


MATH 211 (Calculus III) or equivalent

  1. An understanding of limits, differentiation, and integration.
  2. The ability to evaluate limits, derivatives, and integrals.
  3. Ability to analyze functions and their graphs.
  4. Familiarity with sequences and limits.
  5. Familiarity with logic and proofs

Required Course Materials:

Herbert S. Gaskill and P.P. Narayanaswami, Element of Real Analysis, Prentice Hall 1998

Course Coordinator:

Douglas C. Phillippy, Ph.D., Associate Professor of Mathematics

Course Audience:

Students majoring in mathematics.

Course Objectives:

  1. To develop a rigorous understanding of the real number system and its properties.
  2. To develop a rigorous understanding of real-valued functions with an emphasis on the properties of continuity, differentiability, and integrability.
  3. To develop a rigorous understanding of sequences and series, including sequences and series of real-valued functions.
  4. To introduce students to graduate-level mathematics.
  5. To develop an ability to state and prove theorems.
  6. To enhance the learning process by using technology as a tool to reinforce, expedite, and display mathematical concepts.


    1. The real number system: a) algebraic properties, b) order properties, c) completeness, d) intervals and cluster points, and e) Topology including open and closed sets.
    2. Sequences: a) the limit of a sequence, b) Monotone sequences, c) the Bolzano- Weierstrass Theorem, d) the Cauchy criterion, e) Sequences of functions
    3. Limits and continuity: a) the limit of a function, b) continuous functions, c) continuity on intervals, d) uniform continuity, and e) inverse functions, f) consequences of continuity.
    4. Differentiation: a) the Mean Value Theorem, b) L’Hospital’s rule, and c) Taylor’s Theorem.
    5. Integration: a) the Riemann integral, b) the Fundamental Theorem of Calculus, and c) The integral as a limit.


Revised: May 2012

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