# Topics for Math 3210

The course had seven lectures remaining when the strike began.
Three lectures were given after the end of the strike.

The following topics were covered before the strike:

• Sequences
• Basics and monotone sequences (review): 3.1-3.3
• Cauchy sequences: 3.5
• Subsequences and Bolzano-Weierstrass: 3.4
• Limsup and Liminf.
• Continuity
• Basics (review): 5.1-5.2
• Properties, via sequences: 5.3
• Uniform continuity: 5.4
• Topology
• Open and Closed sets: 10.1
• Calculus
• Derivatives and MVT (Review): 6.1-6.2
• Taylor's theorem: 6.4
• Plus a few miscellaneous topics (eg, iteration of functions, fractals, and - in the book - some results pulled from 7.1, 9.1, and 10.3)
Recall that we worked both in R and in R^d. You should know the basic definitions (convergence, Cauchy, limsup and liminf, cluster points, open and closed sets, closure-interior-boundary, continuity, uniform continuity, differentiability, pointwise and uniform convergence). You should be able to use the definitions, and produce examples and counterexamples to statements involving them. You should understand the main theorems (including hypotheses), and be able to use them. Some of these are: triangle inequality, negation of quantifiers, convergence of monotone bounded sequences, convergence of Cauchy sequences, fixed points of contractions, Bolzano-Weierstrass, Intermediate value thm, Max/Min thm, continuous functions on a closed bounded set are uniformly continuous, MVT and Taylor's thm, and that a uniform limit of continuous functions is continuous. You don't need to remember the proofs, unless I have specifically told you otherwise. Some of the questions will test how well you understand the concepts/results of the course. Others will test your ability to use our techniques to derive new results. Of the topics we've done, the only ones I can think of that you specifically aren't responsible for are Riemann integration (ie, that continuous functions are integrable), and fractals

I had planned to treat power series in some detail, but there wasn't time. The main results we didn't study were:

• Radii of convergence for power series.
• Uniform convergence of partial sums of power series.
• Term-by-term differentiation and integration of power series.
• As a consequence: Every power series is a Taylor series
The main topic covered since the strike was uniform convergence. We studied the definition, and worked out various examples, which illustrated the difference between uniform and pointwise convergence. The main theorem was that a uniform limit of continuous functions is continuous. We proved this theorem, and spent some time understanding it as an example of a general type of "interchange of limits" result. Essentially this was section 8.1 and Theorem 8.2.2 of section 8.2.

Assignment 6 deals with this material, and can be found on the assignments page. It is not to be handed in, but an answer sheet is available. To be prepared for the exam, you should try the the problems yourself (and write them up thoroughly), before looking at the solution sheet.

The exam will have the same format as the midterms, but will be twice as long. You should know the major definitions and theorems from class, and how to use them. In addition, to test your ability to write down correctly a coherent proof, I will ask you to prove one of the following two theorems on the exam:

• A continuous function on a closed bounded interval is uniformly continuous.
• A uniform limit of continuous functions is continuous.