Math Exam Solutions

Exam — Solutions Name: Complete at least five of the following six problems. Extra work will give you extra credit. Show all your work. Closed books and notes, calculators are ok. If you find spelling mistakes you may keep them. Good luck. 1. (20 pts.) Consider the set {0, 1, 12 , 13 , . . . }. Show that this set is compact. Solution 1: Let {Oα } be an arbitrary open cover of A. Then there exists an open set O0 ∈ {Oα } such that 0 ∈ O0 . Since this set is open there exists a δ > 0 such thst (0 − δ, 0 + δ) ⊂ O0 . For this δ there exists a N such that n1 ∈ (0 − δ, 0 + δ) ⊂ O0 for all n ≥ N . Next for n = 1, . . . , N1 let On ∈ {Oα } be an open set such that 1 n ∈ On . Then A ⊂ O0 ∪ O1 ∪ · · · ∪ ON −1 , and {O0 , . . . , On−1 } form a finite subcover of the original open cover. Hence, A is compact. Solution 2: A ⊂ [0, 1] and therefore A is bounded. Furthermaore, all elements of A are boundary points of A, and 0 is the only limit point of A. Therefore A is closed. By the Heine-Borel Theorem A is compact. 2. (20 pts.) Consider the set Q of all rational numbers. Prove that the interior of Q is empty, and that the closure of Q is R. Solution: For every x ∈ R and every δ > 0 the interval (x − δ, x + δ) contains both rational and irrational numbers. This implies that x cannot be an interior point of Q, and that x is a boundary point of Q. Therefore the interior of Q is empty and the boundary of Q is R. Thus Q = b(Q) ∪ Q = R ∪ Q = R. 3. (20 pts.) Let f (x) = ( 0, x, x∈Q x∈ /Q Prove that f is continuous at 0 and nowhere else. Solution: To show that f is continuous at 0 observe that |x| ≥ |f (x)| ≥ 0 and by the squeeze theorem limx→0 f (x) = 0 = f (0). Thus, f is continuous at 0. Now let x 6= 0 if x ∈ / Q there exists a sequence of rational numbers {qn } such that qn → x. Then f (qn ) → 0 6= x = f (x). If x ∈ Q there is a sequence of irrational numbers {pn } such that pn → x, Then limn→∞ pn 6= 0 = f (x). Thus f is not continuous at any x 6= 0. 4. (20 pts.) Let A be a compact set and B a closed set. Show that A ∩ B is compact. Solution: A is compact and therefore closed and bounded. A ∩ B ⊂ A and is also bounded. A ∩ B is the intersection of two closed sets and therefore closed. So A ∩ B is closed and bounded and therefore compact. √ 5. (20 pts.) Use the definition of continuity to show that f (x) = x is continuous on [0, ∞). Is this function uniformly continuous on this set. Justify your answer. √ Solution: Let  > 0, aand x0 be given. Choose δ = x0  then √ √  x0 √ |x − x0 | |x − x0 | | x − x0 | = √ < √ = , √ ≤ √ x0 x0 x + x0 for all x ∈ (x0 − δ, x0 + δ) ∩ [0, ∞) Therefore f is continuous on [0, ∞). Now f is also uniformly continuous on [0, ∞). To see this, we know that zf is uniformly continuous on [0, 1], since this set is compact. On [1, ∞) we may use δ =  for all x0 . Thus f is uniformly continuous on [1, ∞). Now for  > 0 let δ1 be the number we obtain from the uniform continuity of f on [0, 1] and δ2 =  (which works on [1, ∞). Then δ = min{δ1 , δ2 } works for all x0 ∈ [0, ∞) and thus f is uniformly continuous. 6. (20 pts.) Let f : R → R be a continuous function and let let U ⊂ R be an open set. Show that f −1 (U ) is an open subset of R. Solution: Let x0 ∈ f −1 (U ). We need to show that there is a δ > 0 such that (x0 − δ, x0 + δ) ⊂ f −1 (U ). To do this observe that f (x0 ) ∈ U and since U is open there exists an  > 0 such that (f (x0 ) − , f (x0 ) + ) ⊂ U . Since f is continuous there exists a δ > 0 such that f (x) ∈ (f (x0 ) − , f (x0 ) + ) for all x ∈ (x0 − δ, x0 + δ). Therefore, (x0 − δ, x0 + δ) ⊂ f −1 [(f (x0 ) − , f (x0 ) + )] ⊂ f −1 (U ). Hence, f −1 (U ) is open.