MIDTERM SOLUTIONS 2 NOVEMBER 2011 Summary: MIDTERM SOLUTIONS 2 NOVEMBER 2011 1) Suppose (X,d) and (Y,d ) are metric spaces. a) Answer: (p. 36) A function f : X Y is continuous at a X if and only if for every > 0 there is a > 0 so that whenever d(x,a) < , d (f(x), f(a)) < . b) Answer: (p. 44) A function f : X Y is continuous at a X if and only if for every neighborhood V of f(a), the set f-1(V) is a neighborhood of a. 2) a) Answer: (p. 13, combining two definitions on the page) The function g is surjective iff for each z Z there is a y Y so that g(y) = z. b) Suppose f : X Y is also a function. Show that if the composition gf : X Z is surjective, then so is g. Answer: (p. 21 #3) Suppose gf is surjective and z Z. Since gf is surjective there is an x X so that (gf)(x) = z. Let y = f(x) Y. Then g(y) = g(f(x)) = (gf)(x) = z as required. c) Show by clear example that even if gf : X Z is surjective, f is not neces- sarily surjective. Answer: For example, let X = {a},Y = {0,1},Z = {b}. Let f(a) = 0, g(0) = b = g(1). Then gf is clearly surjective (there is only one point in Z) and yet f is not surjective since 1 is not in f(X). 3) Sketch the balls B((-1,2);5) in each of the four metrics described. Collections: Mathematics