Math 3210-1 HW 13 Due Friday, July 13 Properties of Continuous Functions 1. Show that 2x = 3x for some x ∈ (0, 1). 2. Suppose that f : [a, b] → R and g : [a, b] → R are continuous functions such that f (a) ≤ g(a) and f (b) ≥ g(b). Prove that f (c) = g(c) for some c ∈ [a, b]. 3. Suppose that f is a real-valued continuous function on R and that f (a)f (b) < 0 for some a, b ∈ R. Prove that there exists c between a and b such that f (c) = 0. Uniform Continuity 4. Which of the following continuous functions are uniformly continuous on the specified set? Justify your answers. Use any theorems from this section that you wish. (a) f (x) = x17 sin x − ex cos 3x on [0, π] (b) f (x) = x3 on [0, 1] (c) f (x) = x3 on (0, 1) (d) f (x) = x3 on R (e) f (x) = 1 x3 on (0, 1] 5. (a) Prove that if f is uniformly continuous on a bounded set S, then f is a bounded function on S. (b) Use part (a) to prove that x12 is not uniformly continuous on (0, 1). 6. Let g(x) = x2 sin x1 for x 6= 0 and g(0) = 0. (a) Prove that g(x) is continuous on R. (b) Why is g uniformly continuous on any bounded subset of R? (c) Is g uniformly continuous on R?