(a) Consider the function f : R - R defined by 2 if x 0. (i) Prove that f is discontinuous at 0. [3] (ii) Using only the definition of continuity, prove that f is continuous at every c 0. [5] (b) Prove that g: [0, 1] - R defined by g(x) = varcsin(x) is continuous. You may assume without proof that sin: [0, "] - R is continuous and strictly increasing. [3] (c) Let X = (a, b) \\ {c} for some c E (a, b) and let f: X - R be a function. Assume that lim f(x) exists and equals v, and assume that f(x) 2 0 for all x E X. Prove that v 2 0. [5]