MAT4213

From Department of Mathematics at UTSA
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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
5.1

Continuous Functions

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


Week 1/2
5.2

Combinations of Continuous Functions

  • Sums, differences, products and quotients of continuous functions on the same domain
  • Composition of continuous functions
Week 2
5.3

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
5.3

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
5.4

Uniform Continuity

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


Week 3/4
5.4

Lipschitz Functions

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


Week 4
5.4

The Continuous Extension Theorem

  • Uniform continuity and Cauchy sequences
  • The Continuous Extension Theorem


Week 5/6
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
5.6

Inverse Functions

  • The continuous inverse theorem
  • The nth root function


Week 6/7
6.1

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
6.1

Derivatives of Functions with Inverses

  • Relation between continuous, strictly monotone functions and their inverses


Week 7/8
6.2

The Mean Value Theorem

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


Week 8
6.2

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
6.3

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
2.3

Taylor's Theorem

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


Week 9/10
6.4

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
6.4

Newton's Method

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


Week 10/11
7.1

The Riemann Integral

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


Week 11
7.1

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
7.2

Riemann Integrable Functions

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


Week 12
7.2

The Additivity Theorem

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


Week 12/13
7.3

The Fundamental Theorem

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


Week 13
7.3

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
7.3

Integration by Parts

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


Week 14
7.4

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