Due date: Monday, 3/17/14
Last updated Mar 10 17:48 EDT 2014
You are required to do all of the problems below. You will not be required
to hand them all in. I’ve indicated below which ones you do have to hand in.
Don’t make the mistake of thinking that I’m collecting only the problems I
think are important. The “due date” above is the date that your written-up
problems should be handed in, but don’t wait to get started on the assignment.
You should always get started on problems as soon as we cover the relevant
material in class.
- Rosenlicht Chap. VI (pp. 132-135)/ 7, 10, 11, 15, 16 (it’s implicit in the
notation “C([a,b])” that the uniform metric is intended here), 17-20, 22.
Of these, hand in only 7, 10, 11, 15, 16, 20, 22.Note: #20 is not easy. If you have what you think is a quick proof, you
are probably overlooking something, making an implicit assumption, etc.Note on #7: A function f : [a,b] → R (or, more generally, [a,b] → X,
where X is any metric space) is called piecewise continuous if there
exists a partition {xi}i=0N such that (i) the restriction of f to each open
interval (xi-1, xi) is continuous, 1 ≤ i ≤ N, (ii) limx → c– f(x) and limx → c+ f(x)
exist for all c ∈ (a,b), and (iii) limx → a+ f(x) and limx → b– f(x) exist.
(Note that condition (ii) is effectively a statement about the one-sided limits
of f just at the partition points xi, 1 ≤ i ≤ N-1, since condition (i) guarantees
that both one-sided limits exist at all other points of (a,b).) The function f
in problem 7 is not assumed piecewise-continuous; condition (i) is assumed,
but not conditions (ii) and (iii). If you’re able to do this problem only under the
stronger assumption that f is piecewise continuous, that’s still worth something,
but recognize that you’re adding an assumption that’s not stated in the problem.Note on #18: No trig functions or their inverses allowed. The point of the problem
is to show directly that the two integrals are equal to each other, not that they
are equal because they both yield tan–1x. - Read this handout on improper integrals. (This handout was updated Mar. 10.
Notes to the 2009 class have been deleted, and a note to myself in one of the
exercises has been acted upon.) This is good spring-break reading. Exercises
in the handout will be postponed to a later homework assignment.