Last updated Fri Dec 8 17:31 EST 2017
Soft due-date (unenforced, but recommended as a finish-by date): Mon. 12/11/17.
You should do (or at least attempt) all of the problems and reading below. You will not be required to hand in any of the problems, but on the final exam you'll be responsible for the material in this assignment on the final exam.
A: Rosenlicht Chap. IV/ 13, 15–18, 21, 29b, 30
Notes: (1) In #13, there is still more to the proof beyond the last sentence of Rosenlicht's sketch (as well as other details to be filled in earlier). (2) 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. (3) The first part of #17 was done in class, but not the second part. It wouldn't hurt you to do the first part over again anyway. (4) If you do not find #21 harder than the other problems, you are probably doing something wrong. (5) In #30, "closed interval in \(E^2\)" means "closed rectangle \([a,b] \times [c,d]\)" where \(a < b\) and \(c< d\). Hint: first show that the rectangle \(R=[a,b]\times [c,d]\) is arcwise connected, hence connected. Select any point \(p_0\in R=[a,b]\times [c,d]\). As shown in class, \(R\setminus\{p_0\}\) is also arcwise connected, hence connected. Compare the images of \(R\) and \(R\setminus\{p_0\}\) under a continuous function \(f:R\to {\bf R}\).
B: Non-book problem: Use the result of Rosenlicht problem IV/30 to prove the following: a continuous real-valued function on a set \(I\times J\), where \(I\) and \(J\) are (not necessarily closed or bounded) intervals in \({\bf R}\) consisting of more than a single point, cannot be one-to-one. C: Read the handout Continuity, Images, and Inverse Images (also available on the Miscellaneous handouts page).
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