Course Description
MTH 436 is a continuation of MTH 435. We will continue studying the precise mathematical theory behind Calculus.
As in MTH 435, the emphasis will be on mathematical precision and strenthening your skills in proving theorems.
We will again study fundamental concepts of Calculus
that you already know from the intuitive point of view. This time topics include the Riemann integral, sequences of functions, series of constants, series of functions, power series, and the Taylor series.

Many theorems in analysis that we will prove are particular cases of more general theorems
in metric spaces. Therefore, we will continue our study of metric spaces.

Current Downloads

You will find here current downloads: solutions to homework problems, additional class notes etc.

Exams and Evaluation

As in MTH 435, we will have two exams during the semester, Friday March 6, 3-5 and Friday April 17, 3-5.
The final exam will be scheduled by the Office of Enrollment Services based on the time your section meets. The schedule will be posted at:
Final Exam Schedule. According to this grid, our exam is scheduled
for Monday, May 4, 3 pm - 6 pm.

Your grade will be based upon a possible total of 500 points, as
follows:

- Exam 1 -- 100 points,
- Exam 2 -- 100 points,
- Homework -- 150 points,
- Final exam -- 200 points,

Weekly homework will be assigned in class.

Topics

We will begin with few remarks on cardinality, and a few remarks and theorems in complete metric spaces.
Then we will move to Real Analysis. After that we will study more topics in metric space topology.

REAL ANALYSIS

Lipschitzian functions

Relationships between Lipschitzianity, uniform continuity and differentiabilty

Differentiability of monotone functions

Partitions, refinements, the definition of the Riemann integral

Properties of the integral

Sufficient conditions for Riemann integrability

The First Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus

Sequences of functions, pointwise convergence, uniform convergence

Uniform convergence and continuity

Convergence of integrals

Series of constants, convergence

Properties of series, absolute convergence

Convergence tests

Alteranting series, Leibnitz Theorem

Limit inferior, limit superior

Series of functions

Convergence, uniform convergence

Weierstarss M-Test

Series of integrals

Power series, radius of convergence

Taylor polynomials, Taylor series

METRIC SPACES

Open covers, subcovers

Compactness

Sequential compactness

Total boudedness

Characterizations of compactness in metric spaces

Compact subspaces of *R*^{n}

Heine-Borel Theorem

Separable spaces

Continuous Functions

Characterizations of continuity in metric spaces

Continuous mappings on compact spaces