MATH 210 Exam 2, Fall 2013

Math 210 Quiz # 2, 16 November, 2013 1. (a) State 5 important properties of a continuous function f : [a, b] → R. (b) Prove that a continuous function f : K → R, where K is a compact set, attains its supremum at a point in K. 2. Let f : X → Y be a given function, and suppose that f −1(C) is an open subset of X whenever C is an open subset of Y . (a) Prove that f is continuous on X. (b) Prove that f −1(B) is a closed subset of X whenever B is a closed subset of Y . (c) If Y = R, and f is continuous, and a ∈ R, what kind of set is A = {x ∈ X : f(x) ≤ a}? Justify your answer. 3.(a) If f and g are uniformly continuous on a set E, prove that the function f + g is uniformly continuous on ˙E (b) Let h : (0, ∞) → R be the function defined by h(x) = cos 2x + x sin 1 x, 0 < x < ∞. Prove that h is uniformly continuous on (0, ∞). 4. A function f : (a, ∞) → R is differentiable on (a, ∞), and satisfies lim x→∞ (f ′(x) + αf(x)) = 0, where α is a positive constant. (a) Prove that limx→∞ f(x) = 0. (b) Suppose that g is another differentiable function on (a, ∞), that satisfies lim x→∞ (g′(x) + αg(x)) = L, where α > 0, and L are constants. Find limx→∞ g(x), and limx→∞ g′(x) and prove your answers. 5. Let f(x) = 1 − cos x x , 0 < x < π/2. Prove that f is increasing on (0, π/2), and obtain the inequality sin2 x 2 ≤ x π , 0 < x < π/2 . 1