[2+4 +5 = 11 marks] (a) Given a continuous function f RR and a connected subset SR, is f-¹(S) connected? Justify your answer. (b) Given two metric spaces X, p>, and a function f : X → Y that is uniformly continuous on SC X. If a sequence (Tn)neN ES is Cauchy in X, show that (f(n))neN is Cauchy in Y. (c) Given two sequences (fn)neN: (In)neN C C[0, 1] of continuous functions on the closed unit interval [0, 1] defined by nx nx fn(z)= and g(x) = 1+nx²¹ 1+n²x² Find the limit f and g, respectively of each sequence, if it exists. Which of these sequences converge uniformly on [0, 1]? That is, do f and g belong to C[0, 1] or not?