MIDTERM FOR ADVANCED CALCULUS
Instructor: Shu-Yen Pan
Time: 13:10–15:00, Nov. 21, 2001
No credit will be given for an answer without reasoning.
1. [10%]
(i) Is the interval [1, ∞) ⊂ R closed? Why or why not?
(ii) The interval [0, 1) is not compact. Give an open cover of [0, 1) which does not have any finite subcover.
2. [10%] Let f (x) =√ x.
(i) Is f (x) uniformly continuous on [0, 1]? Why or why not?
(ii) Is f (x) uniformly continuous on [0, ∞)? Why or why not?
(iii) Is a bounded continuous function necessarily uniformly continuous? Why or why not?
3. [10%] Let A := { (x, y) ∈ R2| y = sinx1, x 6= 0 } regarded as a subset of R2. (i) Draw the picture of A on the xy-plane.
(ii) What is the interior of A?
(iii) What is the closure of A?
4. [10%] Let a be a real number. Suppose that a 6= 0. Show that a−16= 0 and (a−1)−1= a. (You may use the result: 0 · x = 0 for every x ∈ R2.)
5. [20%] Let d be the standard metric of R2 i.e., d((x1, y1), (x2, y2)) =p
(x1− x2)2+ (y1− y2)2. Define ρ : R2× R2→ R by
ρ((x1, y1), (x2, y2)) = max{|x1− x2|, |y1− y2|}.
(i) Check that (R2, ρ) is a metric space.
(ii) Draw the pictures of the open disk D((1, 1), 1) for (R2, d) and for (R2, ρ) on the xy-plane.
(iii) Show that an open set in (R2, d) is also an open set in (R2, ρ).
6. [10%] Let f : R2→ R defined by (x, y) 7→ x + y. Prove from the definition (i.e., using ²-δ) that f is continuous.
7. [10%] Is it always true that bd(A) = bd(cl(A))? Why or why not?
8. [10%] Let A, B be two compact subsets in a metric space. Show that A ∪ B is also compact.
9. [10%] Suppose that A is a connected subset of R2. Show that the closure of A is also connected.
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