Q3 (25 points) Let S ={f € C[0, 1]: 15 f(x) < 2 for all x € [0, 1]}, S2 ={f € C[0, 1]: f is differentiable on [0, 1] wit

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Q3 25 Points Let S F C 0 1 15 F X 2 For All X 0 1 S2 F C 0 1 F Is Differentiable On 0 1 Wit 1 (29.68 KiB) Viewed 31 times

Q3 (25 points) Let S ={f € C[0, 1]: 15 f(x) < 2 for all x € [0, 1]}, S2 ={f € C[0, 1]: f is differentiable on [0, 1] with f(x) < 0 and 18'(x) < 4 for all x € [0,1]} . (a) As subsets of the metric space (C[0, 1], || . Il sup), which (if any) of S, and Sy are (i) bounded? (ii) open? (iii) closed? (iv) equicontinuous? (v) compact? (b) Show that F(7)(x) = (f(x)) is continuous as a map F: (S1, 11. Ilsup) → (C[0, 1], ||.). (c) is SU S2 connected with respect to the topology induced by || . Ilsup? Justify all your answers. + Drag and drop an image or PDF file or click to browse...