Let Si ={f e C[0, 1]:1 = f(x) < 2 for all x € [0, 1]}, S2 ={f e C[0, 1] : f is differentiable on [0, 1] with f(x) < 0 an

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Let Si F E C 0 1 1 F X 2 For All X 0 1 S2 F E C 0 1 F Is Differentiable On 0 1 With F X 0 An 1 (53.93 KiB) Viewed 20 times

Let Si ={f e C[0, 1]:1 = f(x) < 2 for all x € [0, 1]}, S2 ={f e C[0, 1] : f is differentiable on [0, 1] with f(x) < 0 and 15'(x)] < 4 for all x € [0, 1]}. (a) As subsets of the metric space (C[0, 1], || . || sup), which (if any) of S1 and S2 are x (i) bounded? (ii) open? (iii) closed? (iv) equicontinuous? (v) compact? (b) Show that F(f)(x) = (f(x))2 is continuous as a map F :(S1, || · || sup) → (C[0, 1], || - |li). == (c) Is Si U S2 connected with respect to the topology induced by || . || sup? Justify all your answers.