Last updated Wed Feb 19 19:53 EST 2020
"Virtual" due-date: Tuesday, 2/18/20 (the day before the first midterm).
Due-date for hand-in problems: Monday, 2/24/20.You are required to do all of the problems below. You will not be required to hand them all in. I have indicated below which ones you do have to hand in on the due-date. 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.
A. Rosenlicht Chap. VI/ 1–4, 7, 8, 10, 12, 16, 17. Of these, hand in only 7, 8, 16. Some notes/comments:
- Chapter VI has several of the most challenging problems in the book. At least one of these belongs in the current assignment by virtue of your having been given all the necessary tools, but in view of the upcoming exam I'm postponing it till the next assignment. The current assignment has enough challenging problems already.
- Before you start working on #2, read the Q&A near the bottom of this page.
- It goes without saying that in a yes/no question such as #3, you are expected to prove your answer.
- The "Step-Function Lemma" (Proposition 6.45 in my Notes on Integration) makes problems 2, 3, and 10 easier, but all of them can actually be done without this lemma.
- Regarding #7: A function \(f:[a,b]\to {\bf R}\) (or, more generally, \(f:[a,b]\to X\), where \(X\) is any metric space) is called piecewise continuous if there exists a partition \( \{x_0, \dots, x_N\} \) of \([a,b]\) such that (i) the restriction of \(f\) to the open interval \( (x_{j-1},x_j) \) is continuous, \(1\leq j\leq N\); (ii) \(\lim_{x\to x_j-}f(x)\) and \(\lim_{x\to x_j+}f(x)\) exist, \(1\leq j\leq N-1\); and (iii) \( \lim_{x\to a+} f(x)\) and \( \lim_{x\to b-} f(x)\) exist. The function \(f\) in problem 7 is not assumed piecewise-continuous; condition (i) is assumed, but not conditions (ii) and (iii). For example, the function \(f:[0,1]\to {\bf R}\) defined by \(f(x)=\sin \frac{1}{x}\) for \(x\neq 0\), and \(f(0)=0\), is continuous except at the single point \(0\), but is not piecewise continuous (since \(\lim_{x\to 0+} \sin\frac{1}{x}\) does not exist). 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.
- Problem 12 can, and should, be done without appealing to the ordinary Mean Value Theorem or the Fundamental Theorem of Calculus; the asserted fact is true for much more "primitive" reasons.
- (I [DG] neglected to add #13 to the current assignment when I posted this note, so #13 is NOT part of this assignment. It will be added to the next assignment instead.) To do problem 13 (which is tricky enough to warrant my giving you a hint below), you'll need to use the fact that \(\lim_{n\to\infty} c^{1/n}=1\) for every real number \(c>0\). You may assume this fact. We haven't proven it yet, but it's a quick consequence of something we'll prove in the not-too-distant future, and can also be proven (with some effort) using last semester's homework problem II.11.
Hint for #13: By definition, a sequence \( (A_n)_{n=1}^\infty\) in \({\bf R}\) converges to \(M\in {\bf R}\) if and only if given any \(\epsilon > 0\), we have \(A_n\in (M-\epsilon, M+\epsilon)\) for all \(n\) sufficiently large. In problem 13, for the relevant sequence \(A_n\) and number \(M\), and a given \(\epsilon>0\), the way that you show \(A_n > M-\epsilon\) for all \(n\) sufficiently large will be quite different from the (easier) way that you show \(A_n < M+\epsilon\) for all \(n\) sufficiently large. Continuity of \(f\) is critical to the "\(A_n > M-\epsilon\)" argument, but irrelevant to the "\(A_n < M+\epsilon\)" argument.- In #16, it's implicit in the notation "\(C([a,b])\)" that the uniform metric is intended. Before doing this problem, do Exercise 6.5 in my notes; this is one of the "part C" problems below. (Rosenlicht's #16 is intended to be done after his #9, which I haven't assigned since it's exactly part (b) of my Exercise 6.5.)
B. No non-book problems (yet) other than the ones from my notes ("part C" below).
C. In the Notes on Integration, do exercises 6.1, 6.3–6.8 (located on pp. 6–7, 12, 21, 23, 31, and 32 in the Feb. 14 version of the notes). Of these, hand in only exercise 6.6.
Reminder: Whenever I make a revision that causes a numbering-change or significant pagination-change in material we've already covered, I keep the previous versions of the notes posted on the Miscellaneous Handouts page, in case you're having trouble finding something whose location or numbering has changed. I don't usually change the version-date for minor revisions, such as fixing typos, or for revisions that are further along than wherever we are in the notes.Q&A. Below are a question and answer from a previous time that I taught this class. A similar question can occur in many other problems. The answer is always the same (except, of course, for specific reference to the particular exercise).
Question. "For exercise 2 in Rosenlicht Chapter VI, may we assume that the integral exists and use that assumption to show that it is equal to 0, or do we have to prove existence first and then prove that it equals 0?"
Answer. You absolutely may not assume that the integral exists. Any time you're asked to prove "this = that", and "this" does not automatically exist, it's implicit that what you're being asked to prove is "this exists AND this = that" (etc. for inequalities). Another example of this sort of thing is "Prove that a certain limit equals 2." It should be obvious that if you were asked this, you would not be allowed to assume from the start that the limit exists. If the writer of the exercise wanted you to make such an assumption, he/she would say so explicitly.
However, often in proving "this = that" (or "this \(\geq\) that", etc.), you end up exhibiting the value of "this" (or the limit, in my example) at the same time that you prove existence. For example, you often prove that a limit exists by intelligently guessing the value of the limit (before trying to write a proof), then showing that your function, sequence, or whatever, approaches that value. Very often in such cases, "intelligent guesswork" does involve a step of the type "Hmm, suppose the limit (or whatever) existed. What would it have to be?" That's perfectly good thinking when you're trying to figure out a way to prove that a limit (or whatever) exists. Of course, this will not be part of the proof itself; you're going through this step to guide you to a proof.
General homework page
Class home page