## Math 355: Introduction to Analysis Study Guide for Exam 2

The second midterm exam questions will be posted on Moodle on Monday, April 9 at 12:00pm. A printed copy for you to write on will be given out on Tuesday, April 10, at the end of class. You will have until 5:00pm on Friday, April 13, to complete this take-home exam.

If you would like, you have the option of shifting this schedule by 24 hours to see the exam questions a day later (so the Moodle post will not be available to you until 12:00pm on Tuesday), and then you would hand in the exam 5:00pm on Saturday, April 14. If you prefer this option, please email me as soon as possible, but definitely by 9:00am on Monday April 9th.

A review session, led by Obinna Ukogu, will be held on Sunday, April 8, 7-8pm in Seeley Mudd 207.

The following is a chapter by chapter list of topics intended to help you organize the material we have covered in class as you study for your exam. It is only intended to serve as a guideline, and may not explicitly mention everything that you need to study. While this exam will emphasize the chapters below, it is cumulative, and will therefore also cover material from earlier sections. Please be sure to review homework and example problems for the relevant chapters.

For this exam, you may use the course textbook, your class notes, your old homework sets and exam, and the solutions to these which were posted on the course Moodle page. You may not use any other outside help, including other websites or texts, and you may not discuss the exam or related mathematics with anyone else.

3.3: Know the definitions of compactness, open covers, and subcovers, and the various characterizations and properties given in Theorems 3.3.4, 3.3.5 and 3.3.8.

4.2: Understand how we define the limit of a function, and the properties given in Theorem 4.2.3 (Sequential Criterion for Functional Limits), Corollary 4.2.4 (Algebraic Limit Theorem for Functional Limits), and Corollary 4.2.5 (Divergence Criterion for Functional Limits).

4.3: Know all the different ways we can define function continuity, as given in Theorem 4.3.2. Know how to use Corollary 4.3.3 (Criterion for Discontinuity), Theorem 4.3.4 (Algebraic Continuity Theorem), and Theorem 4.3.9 (Composition of Continuous Functions).

4.4: Be familiar with the behaviour of continuous functions on compact sets. In particular, know that a continuous function preserves compact sets (Theorem 4.4.1) and how this leads to the Extreme Value Theorem (Theorem 4.4.2). Also know what we mean by uniform continuity, the Sequential Criterion for Absence of Uniform Continuity (Theorem 4.4.5) and that a continuous function on a compact set is uniformly continuous (Theorem 4.4.7).

4.5: Know the Intermediate Value Theorem, and the proofs based on Axiom of Completeness and the Nested Interval Property. You do not need to know Theorem 4.5.2.

5.2-5.3: Know the definition of differentiability, and be familiar with the following theorems: Theorem 5.2.3 (Differentiability Implies Continuity), Theorem 5.2.4 (Algebraic Differentiability Theorem), Theorem 5.2.5 (Chain Rule), Theorem 5.2.6 (Interior Extremum Theorem), Theorem 5.3.1 (Rolle's Theorem), Theorem 5.3.2 (Mean Value Theorem), Corollary 5.3.3 (Derivative 0 Implies Constant Function) Theorem 5.3.5 (Generalized Mean Value Theorem), Theorem 5.3.6 (L'Hospital's Rule).

6.2: Understand what it means for a sequence of functions to converge pointwise and to converge uniformly. Know that the limit of a sequence of continuous functions that converges uniformly is continuous (Theorem 6.2.6). It is also important to note that this does not always hold for pointwise convergence (Example 6.2.2.ii). You may also read and use Theorem 6.2.5 (Cauchy Criterion for Uniform Convergence).

The following practice exam is only intended to give you an idea of what the exam will be like. It certainly does not cover everything that might appear on the actual exam, so please make sure you do study all topics discussed in class! However, this should provide you with a sense of the different types of questions you may encounter. (It may also include some pieces for which you already have an explicit answer in your course notes or the homework solutions, but try to solve those specific questions as if you did not already have the answers.)