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	Continuous Functions	The Definition of the Limit of a Function The Sequential Criterion and Divergence Criteria	The definition of a continuous function at a point Sequential criterion for continuity Discontinuity criterion
Week 1/2	5.2	Combinations of Continuous Functions	Continuous Functions Composition of 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	Domain and Range of a Function Intervals Continuous Functions	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	Continuous Functions on Intervals Continuous Functions	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	Continuous Functions Continuous Functions on Intervals	The definition of uniform continuity Nonuniform continuity criteria Uniform continuity theorem
Week 3/4	5.4	Lipschitz Functions	Uniform Continuity Functions(The Cartesian Product Definition)	Definition of a Lipschitz function If a function is Lipschtiz, then it is uniformly continuous
Week 4	5.4	The Continuous Extension Theorem	Continuous Functions on Intervals Uniform Continuity	Uniform continuity and Cauchy sequences The Continuous Extension Theorem
Week 4/5	5.4	Approximations of Continuous Functions	Domain and Range Continuous Functions on Intervals The Continuous Extension Theorem	Definition of a step function On closed and bounded intervals, continuous functions can be approximated by piecewise linear functions Weierstrass approximation theorem
Week 5	5.5	Continuity and Gauges	Suprema, Infima, and the Completeness Property Intervals	Definition of a partition of an interval Definition of a tagged partition of an interval Definition of a gauge of an interval The existence of delta-fine partitions
Week 5/6	5.6	Monotone Functions	Monotone Sequences Suprema, Infima, and the Completeness Property Continuous Functions on Intervals	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	Monotone Functions Continuous Functions on Intervals	The continuous inverse theorem The nth root function
Week 6/7	6.1	The Derivative	Continuous Functions on Intervals The Definition of the Limit of a Function	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	Inverse Functions The Derivative Combinations of Continuous Functions	Relation between continuous, strictly monotone functions and their inverses
Week 7/8	6.2	The Mean Value Theorem	The Derivative Continuous Functions on Intervals	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	Continuous Functions on Intervals The Mean Value Theorem	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	The Mean Value Theorem The Limit Theorems for Functions	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	Intro to Polynomial Functions The Derivative The Mean Value Theorem	Taylor's Theorem Applications of Taylor's theorem
Week 9/10	6.4	Relative Extrema and Convex Functions	The Derivative Taylor's Theorem	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	The Derivative Taylor's Theorem	Description of Newton's Method Examples of Newton's Method
Week 10/11	7.1	The Riemann Integral	The Definition of the Limit of a Function Continuity and Gauges	Partitions and tagged partitions The definition of the Reimann integral
Week 11	7.1	Properties of the Integral	The Riemann Integral Bounded Functions	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 Riemann Integral Properties of the Integral	The Cauchy Criterion The squeeze theorem for integrals functions A step function is Riemann integrable
Week 12	7.2	The Additivity Theorem	The Riemann Integral Riemann Integrable Functions	The Additivity Theorem Interchanging the upper and lower bounds of the Riemann integral
Week 12/13	7.3	The Fundamental Theorem	Continuous Functions The Riemann Integral	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	Continuous Functions The Riemann Integral	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 Fundamental Theorem The Substitution and Composition Theorems	The method of Integration by Parts Taylor's Theorem with Remainder
Week 14	7.4	The Darboux Integral	Suprema, Infima, and the Completeness Property Partitions Bounded Functions	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
Week 15	7.5	Approximate Integration	The Fundamental Theorem Taylor's Theorem The Riemann Integral	Equal Partitions The trapezoidal Rule The midpoint Rule Simpson's Rule