Math 104 homework 6 | Mathematics homework help

HOMEWORK ASSIGNMENT 6 FOR MATH 104 LECTURE 002, SUMMER 2014 JASON FERGUSON [email protected] DUE: 6pm Thursday, July 17th in class or 6:30pm Thursday, July 17th by e-mail. Hints are encoded using ROT13; i.e. A↔N, B↔O, C↔P, . . . , M↔Z. To decode them quickly, copy and paste the text to http://www.rot13.com/ and press “cypher”. I did this so that people who don’t want hints won’t have them spoiled, but people who do want hints can read them quickly. Continuity 1. Let f : R → R be the function defined as follows: If x is irrational then f(x) = 0. If x is rational, and we write x as a fraction in lowest terms as p q with p ∈ Z and q ∈ N, then f(x) = 1 q . Thus, f(π) = 0, f(−2.2) = f −11 5 = 1 5 , and f(0) = f 0 1 = 1 1 = 1. Prove that f is continuous at every irrational number, but f is discontinuous at every rational number. Pugh calls f the “rational ruler function” and includes a paragraph on it on p.161, and a sketch of it on p.162. Closure, Interior, Boundary 2. Let M be any metric space and A and B any subsets of M. Prove: a. int A = M M A b. A ∪ B = A ∪ B c. A is closed if and only if A = A d. A = A. e. If A ⊂ B then A ⊂ B f. Say what the versions of b–e are for interiors instead of closure. Hint 1: Nyy cnegf bs guvf fubhyq or fubeg. Hint 2: Hfr gur fznyyrfg pybfrq frg qrsvavgvba bs pybfher. 3. Let M be any metric space, let p ∈ M and > 0 be arbitrary. a. Show that M(p) ⊆ {q ∈ M : d(p, q) ≤ }. b. Show that if M = R 2 [using the Euclidean distance], that M(p) = {q ∈ M : d(p, q) ≤ }. [You should be able to generalize your proof to R n for any n. In fact, the same statement and proof should work if M is any real or complex vector space and d is defined as d(v, w) = kv − wk for some norm on M, but you don’t need to show it in this generality.] c. Give a specific example of a metric space M, a point p ∈ M, and a positive real number for which M(p) 6= {q ∈ M : d(p, q) ≤ }. Hint for a: Hfr gur erfhyg bs Ceboyrz Guerr sebz Ubzrjbex Nffvtazrag Svir. Hint for b: Svefg gel gb cebir vg va gur fcrpvsvp pnfr jura c vf (0,0) naq rcfvyba vf bar. Hint for c: Hfr gur erfhyg bs Ceboyrz Fvk-N sebz Ubzrjbex Nffvtazrag Svir. Subspaces 4. Show that any function that is obtained by restricting the domain and codomain of a continuous function is itself continuous. Hint: Hfr Pbebyynel Fvkgrra sebz Chtu, naq gur bcra frg qrsvavgvba bs pbagvahvgl. 5. Gluing. A common way of defining a function f : X → Y is to define them piecewise; i.e. by gluing other functions together: You pick subsets {Xi} of X whose union is X, then give a function fi : Xi → Y for each i. You show that for any i, j and any x ∈ Xi ∩ Xj that fi(x) = fj (x). Then you let f(x) = fi(x) on Xi for all i. As a specific example: Let f1 : (−∞, 1] → R be f1(x) = 3x − 5 and f2 : [1, ∞) → R be f(x) = x − 3. Then since f1(1) = −2 = f2(1), we can define f : R → R as: f(x) = ( 3x − 5 if x ∈ (−∞, 1] x − 3 if x ∈ [1, ∞) . 1 a. Let M and N be any metric spaces, and let {Mi} be any open sets in M whose union is M. Suppose that for each i we have a continuous function fi : Mi → N [where Mi is regarded as a subspace of M] and that they agree on overlaps, i.e. fi(x) = fj (x) for all x ∈ Mi ∩ Mj . Then define f : M → N by letting f(x) = fi(x) if x ∈ Mi . Show that f is continuous. b. Same setup and conclusion as (a), except now all of the {Mi} are closed and there are only finitely many of them. [Notice that this can’t work for infinitely many closed {Mi} because we could then just pick each Mi to be a one-point set, and then make any function f : M → N out of it. That means you need to use finiteness somewhere.] Hint for a: Hfr Pbebyynel 16 sebz Chtu, naq gur bcra frg qrsvavgvba bs pbagvahvgl. 6. At the bottom of p.67 Pugh claims without any explanation that the set S = x ∈ Q : − √ 2 < x < √ 2 is both open and closed as a subset of Q, but is neither open nor closed as a subset of R. Prove this. EDIT: “Prove this” means “Prove that S is both open and closed in Q and is neither in R” and not “Prove that Pugh makes this claim on p.67 without any explanation.” Possible hint: Hfr gur erfhyg bs Ceboyrz Guerr sebz Ubzrjbex Nffvtazrag Svir. Product Spaces 7a. Let M be any metric space and (xn)∞ n=1 and (yn)∞ n=1 be any two sequences in M that converge to x ∈ M and y ∈ M, respectively. Show that the sequence of real numbers d(xn, yn) ∞ n=1 converges to d(x, y). b. Explain why 7a shows that d: M × M → R is continuous. Hint for a: q(k,l) vf yrff guna be rdhny gb q(k,ka) cyhf q(ka,la) cyhf q(la,l). Yvxrjvfr, q(ka,la) vf yrff guna be rdhny gb q(ka,k) cyhf q(k,l) cyhf q(l,la). Hfr gurfr vardhnyvgvrf gb rfgvzngr q(k,l) zvahf q(ka,la). 8a. How much time did you spend on this assignment? b. Do you think this assignment was too long? Too short? Just right? c. Do you think that the individual questions were too easy? Too hard? Just right? 2

Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
$26
The price is based on these factors:
Academic level
Number of pages
Urgency
Basic features
  • Free title page and bibliography
  • Unlimited revisions
  • Plagiarism-free guarantee
  • Money-back guarantee
  • 24/7 support
On-demand options
  • Writer’s samples
  • Part-by-part delivery
  • Overnight delivery
  • Copies of used sources
  • Expert Proofreading
Paper format
  • 275 words per page
  • 12 pt Arial/Times New Roman
  • Double line spacing
  • Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read more

Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read more

Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read more

Privacy policy

Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read more

Fair-cooperation guarantee

By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more

Order your essay today and save 15% with the discount code GRADUATE