From Department of Mathematics at UTSA
Jump to navigation Jump to search

The textbook for this course is Introduction to Real Analysis by Bartle and Sherbert

A comprehensive list of all undergraduate math courses at UTSA can be found here.

The Wikipedia summary of Real Analysis.

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1

Continuous Functions

  • The definition of a continuous function at a point
  • Sequential criterion for continuity
  • Discontinuity criterion

Week 1/2

Combinations of Continuous Functions

  • Sums, differences, products and quotients of continuous functions on the same domain
  • Composition of continuous functions
Week 2

Continuous Functions on Intervals

  • Bounded Functions
  • The boundedness theorem on closed and bounded intervals
  • Definitions of absolute maximum and absolute minimum of a function
  • The maximum-minimum theorem

Week 2/3

The Intermediate Value Theorem

  • The Location of Roots theorem
  • The Intermediate Value Theorem
  • The image of a continuous functions on a closed and bounded interval is a closed and bounded interval

Week 3

Uniform Continuity

  • The definition of uniform continuity
  • Nonuniform continuity criteria
  • Uniform continuity theorem

Week 3/4

Lipschitz Functions

  • Definition of a Lipschitz function
  • If a function is Lipschtiz, then it is uniformly continuous

Week 4

The Continuous Extension Theorem

  • Uniform continuity and Cauchy sequences
  • The Continuous Extension Theorem

Week 5/6

Monotone Functions

  • Monotone functions
  • The left and right hand limits for interior points of monotone functions
  • Defining the jump pf a function at a point
  • For a monotone function on an interval, the set of points at which the function is discontinuous is countable

Week 6

Inverse Functions

  • The continuous inverse theorem
  • The nth root function

Week 6/7

The Derivative

  • Definition of the derivative of a function at a point
  • Continuity is required for a function to be differentiable
  • The constant, sum, product and quotient rules for derivatives
  • Caratheodory's theorem
  • The chain rule

Week 7

Derivatives of Functions with Inverses

  • Relation between continuous, strictly monotone functions and their inverses

Week 7/8

The Mean Value Theorem

  • Relative maximum and relative minimum of a function
  • Interior extremum theorem
  • Rolle's Theorem
  • The Mean Value theorem

Week 8

Extrema of a Function

  • Increasing and Decreasing functions
  • First Derivative test for extrema
  • Applications of the Mean Value Theorem
  • The Intermediate Value Property of Derivatives

Week 8/9

L'Hospital's Rules

  • Intermediate forms
  • The Cauchy mean value theorem
  • L'Hospital's Rule for limits of the 0/0 form
  • L'Hospital's Rule for limits with infinity in the denominator
Week 9

Taylor's Theorem

  • Taylor's Theorem
  • Applications of Taylor's theorem

Week 9/10

Relative Extrema and Convex Functions

  • Using higher order derivatives to determine where a function has a relative maximum or minimum
  • Definition of a convex function
  • Determining whether a function is convex using the second derivative

Week 10

Newton's Method

  • Description of Newton's Method
  • Examples of Newton's Method

Week 10/11

The Riemann Integral

  • Partitions and tagged partitions
  • The definition of the Reimann integral

Week 11

Properties of the Integral

  • Integrals multiplied by constants
  • The sum of two integrals on a common interval
  • If the functions f is less than the function g on some interval, then the integral of f will be less than the integral of g on that same interval.
  • The Boundedness Theorem

Week 11/12

Riemann Integrable Functions

  • The Cauchy Criterion
  • The squeeze theorem for integrals functions
  • A step function is Riemann integrable

Week 12

The Additivity Theorem

  • The Additivity Theorem
  • Interchanging the upper and lower bounds of the Riemann integral

Week 12/13

The Fundamental Theorem

  • Part one of the Fundamental theorem of calculus
  • Part two of the fundamental theorem of calculus

Week 13

The Substitution and Composition Theorems

  • The substitution theorem
  • Examples of evaluating integrals using the change of variable method
  • The Composition Theorem
  • The Product Theorem

Week 13/14

Integration by Parts

  • The method of Integration by Parts
  • Taylor's Theorem with Remainder

Week 14

The Darboux Integral

  • Upper and Lower sums
  • Upper and Lower integrals
  • The Darboux Integral
  • If a function is either continuous or monotone on a closed interval, then it is Darboux integrable on that interval.
  • The equivalence of the Riemann and the Darboux integrals