Last updated Mon Dec 2 23:34 EST 2019
"Virtual due-date": Friday 12/6/19, but the due-dates for reading are earlier.
You are required to do all of the problems below. You will not be required to hand in any of them.
C:
- Read the second Proposition on p. 60 of Rosenlicht (both the statement and the proof). Due-date Monday 12/2/19.
- Read Sections IV.1–IV.3 of Rosenlicht. Due-date Monday 12/2/19.
- Read the handout Continuity, Images, and Inverse Images. Tentative due-date: Wed. 12/4/19; will not be earlier than this.
A: Rosenlicht Chap. IV/ 1–4, 6, 7, 16.
Notes: (1) "Discuss the continuity of ..." means: Determine, with proof, at which points the indicated function is/isn't continuous. If you see how to apply problem #3 to help with some proof(s) in problem #1, you may do so. (2) In problem 1b, among the "general properties of the sine function" you're allowed to assume is continuity. (2) Problem #7 can be done from scratch, but can be done more efficiently by using the result of #3 (once you've done #3). (3) In #16, to use Rosenlicht's hint you have to prove that both halves of the hint are true. Note that in general a continuous function \(f\) from one metric space to another need not carry closed sets to closed sets. (For an example, see the handout in part C of this assignment, Example 2a.) To use Rosenlicht's hint, part of what you have to figure out is why the \(f\) in this problem does carry closed sets to closed sets.B: Click here for non-book problems.
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