Math 104, Section 6 Fall 2003SarasonREVIEW PROBLEMS1. Let X and Y be metric spaces. Prove that the function f : X → Y is continuous if and onlyif f(¯A) ⊂f(A) for every subset A of X.2. For A and B subsets of R, let A + B = {a + b : a ∈ A, b ∈ B}.(a) Prove that if A is open then A + B is open.(b) Prove that if A and B are compact then A + B is compact.(c) Find an example of two closed sets A and B such that A + B is not closed.3. (a) Prove that if U1and U2are open subsets of Rkthen U1× U2is an open subset of R2k.(b) Prove that if F1and F2are closed subsets of Rkthen F1× F2is a closed subset of R2k.4. Let U be an open subset of a metric space X. Let A beasubset of U whose limit points areall in X\U. Prove U\A is open.5. Let F be a subset of Rkwith the property that F ∩ K is compact whenever K is a compactsubset of Rk. Prove F is closed.6. Find a countable subset of R2whose set of limit points is {(x, y) ∈ R2: x2+ y2=1}.7. Let A be an uncountable subset of R. Prove that the set A\Ais at most countable.8. Let X beametric space and f : X → X a continuous function. Prove {x ∈ X : f (x)=x},the set of fixed points of f,isclosed.9. Let the function f :(0, 1) → R be uniformly continuous. Prove that limx→0f(x) exists.10. Let X beametric space. Let f : X → R and g : X → R be continuous functions, and definethe function h : X → R by h(x)=max{f(x),g(x)}. Prove h is continuous.11. Let the function f : R → R be continuous and periodic with period 1 (i.e., f(x +1)=f(x)for all x). Prove f is uniformly continuous.12. Let the function f : R → R be continuous and satisfy lim|x|→∞f(x)=0. Prove f is uniformlycontinuous.13. (a) Let the function f :(a, b) → R be nondecreasing. Prove that the set of discontinuitiesof f is at most countable.(b) Let A be a countable subset of (0, 1). Construct a nondecreasing function f :(0, 1) → Rwhose set of discontinuities is A.14. Let X be a compact metric space, Y a metric space, and f : X → Y a one-to-one continuousfunction of X onto Y . Prove that the inverse function f−1is continuous.15. Let c be the space of convergent sequence s =(sn)∞1, with the usual metric. Define thefunction f : c → R by f(s)= limn→∞sn. Prove f is continuous.16. Let c be the space of convergent sequences s =(sn)∞1. Define the function f : c → cby f(s)=s1+ ···+ snn∞1. Prove f is continuous. (That f maps c into c is given byHomework Exercise F.)17. Prove that every point in [0, 2] is the sum of two numbers in the Cantor set.18. Let K be a compact set in a metric space X, and let U be an open cover of K. Prove there isapositive number δ such that any two points x and y in K satisfying d(x, y) <δlie togetherin some set of the cover U. (Lebesgue covering lemma)19. Let X be a compact metric space and f : X → X a continuous function. Prove there is anonempty compact subset K of X such that f(K)=K.20. Let X and Y be metric spaces. The graph of a function f : X → Y is the subset Gf={(x, f(x)) : x ∈ X} of X ×Y . Regard X × Y as a metric space with the metricdX×Y((x1,y1), (x2,y2)) = dX(x1,x2)+dY(y1,y2).(a) Prove that if f is continuous then Gfis closed.(b) Find an example of a discontinuous function f such that Gfis closed.(c) Prove that if Y is compact and Gfis closed, then f is