Reference no: EM131127907

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 ε Z and q ε 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_ε(p))^- ⊆ {q ε M : d(p, q) ≤ ε}.

b. Show that if M = R² [using the Euclidean distance], that

Mε(p) = {q ε M : d(p, q) ≤ ε}.

[You should be able to generalize your proof to Rⁿ 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

Mε(p) = {q ε M : d(p, q) ≤ε }.

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, ∞)

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?

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