- Q3 25 Points Let Si Fe C 0 1 1 F 2 2 For All I E 0 1 S2 Fe C 0 1 F Is Differentiable On 0 1 With F X 1 (33.68 KiB) Viewed 19 times
Q3 (25 points) Let Si={fe C[0,1]:1<f(2) <2 for all I e [0,1]}, S2 ={fe C[0,1] : f is differentiable on [0, 1] with f(x) < 0 and|f'(2) <4 for all 2 € [0,1]}. (a) As subsets of the metric space (C[0, 1], || . || sup), which (if any) of S, and S, are (0) bounded? (ii) open? (iii) closed? (iv) equicontinuous? (v) compact? (b) Show that F(f) (T) = (f(t)) is continuous as a map F: (S., || - || sup) + (C[0,1], || . ||1). ©) Is SU S2 connected with respect to the topology induced by || . ||sup? Justify all your answers.