(a) (i) Explain why there can be no continuous function f [0, 1] (0,1) that...

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(a) (i) Explain why there can be no continuous function f [0, 1] → (0,1) that is a surjection. (ii) Give an example of a continuous surjection g: (0, 1)→ [0, 1]. (b) (i) (ii) [5] State the Mean Value Theorem. Let f: RR be a differentiable function, whose derivative f' is continuous. Let a ER be a point for which f(a) = 0 but f'(a) 0. Use the Mean Value Theorem to show that there exists > 0 such that f(x) #0 whenever 0 < x-a <8. [10] (a) (i) Explain why there can be no continuous function f [0, 1] → (0,1) that is a surjection. (ii) Give an example of a continuous surjection g: (0, 1)→ [0, 1]. (b) (i) (ii) [5] State the Mean Value Theorem. Let f: RR be a differentiable function, whose derivative f' is continuous. Let a ER be a point for which f(a) = 0 but f'(a) 0. Use the Mean Value Theorem to show that there exists > 0 such that f(x) #0 whenever 0 < x-a <8. [10]

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a Question 1 There can be no continuous function f 0 1 0 1 because 0 1 is not a closed set A surject... View the full answer