# MAT4213

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

Week 1/2
5.2
• Sums, differences, products and quotients of continuous functions on the same domain
• Composition of continuous functions
Week 2
5.3
• 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 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
• The definition of uniform continuity
• Nonuniform continuity criteria
• Uniform continuity theorem

Week 3/4
5.4
• Definition of a Lipschitz function
• If a function is Lipschtiz, then it is uniformly continuous

Week 4
5.4
• Uniform continuity and Cauchy sequences
• The Continuous Extension Theorem

Week 5/6
5.6
• 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
• The continuous inverse theorem
• The nth root function

Week 6/7
6.1
• 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
• Relation between continuous, strictly monotone functions and their inverses

Week 7/8
6.2
• Relative maximum and relative minimum of a function
• Interior extremum theorem
• Rolle's Theorem
• The Mean Value theorem

Week 8
6.2
• 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
• 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
• Applications of Taylor's theorem

Week 9/10
6.4
• 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
• Description of Newton's Method
• Examples of Newton's Method

Week 10/11
7.1
• Partitions and tagged partitions
• The definition of the Reimann integral

Week 11
7.1
• 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
• The Cauchy Criterion
• The squeeze theorem for integrals functions
• A step function is Riemann integrable

Week 12
7.2
• Interchanging the upper and lower bounds of the Riemann integral

Week 12/13
7.3
• Part one of the Fundamental theorem of calculus
• Part two of the fundamental theorem of calculus

Week 13
7.3
• The substitution theorem
• Examples of evaluating integrals using the change of variable method
• The Composition Theorem
• The Product Theorem

Week 13/14
7.3
• The method of Integration by Parts
• Taylor's Theorem with Remainder

Week 14
7.4
• 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