Due-date (unenforced): Friday, 12/12/14.
(There are no hand-in problems, and classes end on Wednesday. But try to do all the problems by the date above, so that you have time to ask me questions and digest my answers.)
Last update made by D. Groisser Thurs Dec 11 16:44 EST 2014
- A: Rosenlicht pp. 90–95/ 9, 10, 13-19, 21, 29b, 30.
Notes: (1) 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. To use the hint, part of what you have to figure out is why the f in this problem does carry closed sets to closed sets. (2) In #30, “closed interval in E2” means “closed rectangle [a,b] × [c,d]” where a<b, c<d . Hint: first show that (rectangle)\{point} is arcwise connected, hence connected (by the result of #29a, which will be proven in class on Wednesday, or which you can prove on your own first). (3) The result of #30 implies that a continuous real-valued function on a non-necessarily-closed rectangle I×J, where I and J are intervals in R consisting of more than a single point, cannot be one-to-one. How?
- B: Click here for non-book problems.
- C: Read the handout Continuity, Images, and Inverse Images (also available on the Miscellaneous handouts page).
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