# Analysis- continuity and differentiability

Hi, could somebody please help me with the following question, I have been stuck on it for ages.

1. let f[0,1] -> R be continuous with f(0)=0, f(1)=1. Prove the following:

a.(i) If for c in (0,1) f is differentiable at c with f'(c)<0 then there are exists points y such that f(x)=y has more than one solution.

(ii). If f(x)>0 for x>0 show there is $$\delta$$ in (0,1) with $$\delta$$ not equal to f(x) for all x in [0.5,1].

(iii). If f is also differentiable on (0,1) prove there is an a in (0,1) such that

f'(a)/cos(a$$\pi/2$$) = $$2/\pi$$

b. If f:[-1,1]->R is such that f(sin(x)) is continuous on R then is f continuous. (give a proof or counterexample).

3. I am really stuck on this question and so have not got very far at all. I think part 1 must involve the application of the intermediate value theorem twice though I can't see how to do so. I don't even know how to start part 2. For part 3 I think I will need to use the mean value theorem (and probably a previous question) but can't see where they have got the cos function from.For part b I suspect that it is true but don't know how to write a proof.

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