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Introduction to Real Analysis

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Prerequisites: Qualifying math score and successful completion of Introduction to Abstract Math or the equivalent.

Course Format: Individually Paced

Course Length: Typically 6 months

Recommended School Credit: One full year of high school credit or one semester of college credit equal to or greater than an AP class

Course Code: IRA

Course Description


Introduction to Real Analysis gives advanced students the theoretical foundations underlying the topics taught in a typical Calculus AB and BC course. Introduction to Real Analysis will cover algebraic and order properties of the real numbers, the least upper bound axiom, limits, continuity, differentiation, the Riemann integral, sequences, and series. Definitions and proofs will be stressed throughout the course. Online course materials supplement the required textbook.

Each student is assigned to a CTY instructor to help them during their course. Students can contact their instructor via email with any questions or concerns at any time. Live one-on-one online review sessions can be scheduled as well to prepare for the graded assessments, which include homework, chapter exams, and a cumulative midterm and final. Instructors use virtual classroom software allowing video, voice, text, screen sharing and whiteboard interaction.

Topics include:

  • The Real Numbers
  • Sequences
  • Functions and Continuity
  • The Derivative
  • The Integral
  • Infinite Series
  • Sequences and Series of Functions
  • Introduction to Differential Equations

See the List of Topics tab for the complete listing of topics covered.

sample math problem

Materials Needed

A textbook purchase is required for this course:

An Introduction to Analysis, 2nd Ed, by Bilodeau, Thie, Keough

ISBN: 978-0-7637-7492-9

Introduction to Analysis textbook cover

List of Topics

Upon successful completion of the course, students will be able to demonstrate mastery over the following topics:

The Real Numbers

  • Sets
  • Functions
  • Algebraic and order properties
  • The positive Integers
  • The least upper bound axiom


  • Sequences and limits
  • Limit theorems
  • Monotonic sequences
  • Subsequences defined inductively
  • Subsequences
  • Cauchy sequences
  • Infinite Limits

Functions and Continuity

  • Limit of a function
  • Limit theorems
  • Other limits
  • Continuity
  • Intermediate values, extreme values
  • Uniform continuity
  • Function of two variables

The Derivative

  • The definition of the derivative
  • Rules for differentiation
  • The Mean Value Theorem
  • Inverse functions
  • Differentiability in R2

The Integral

  • The definition of the integral
  • Properties of the integral
  • Existence theory
  • The Fundamental Theorem of Calculus
  • Improper integrals
  • Double integrals

Infinite Series

  • Basic theory
  • Absolute convergence
  • Power series
  • Taylor series

Sequences and Series of Functions

  • Uniform convergence
  • Consequences of uniform convergence
  • Two examples

Introduction to Differential Equations

  • Elementary first order differential equations
  • Existence and uniqueness
  • Power series solutions

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Technical Requirements

This course requires a properly maintained computer with high-speed internet access and an up-to-date web browser (such as Chrome or Firefox). The student must be able to communicate with the instructor via email. Visit the Technical Requirements and Support page for more details.

This course uses an online classroom for individual or group discussions with the instructor. The classroom works on standard computers with the Adobe Flash plugin, and also tablets or handhelds that support the Adobe Connect Mobile app.