Could somebody please help me with the following?
Let f be a real continuous function. Which of the following statements are true and which are false? Give proofs or counter examples to justify your answers.
1. If f is a bijection and is also differentiable then its inverse is differentiable.
I don't know how to start this part of the question, any ideas?
3. If f is differentiable with f'(x) not equal to zero for any x, then f is a bijection.
I think this is true, I have used Rolles theorem to prove it must be injective but cannot prove it is surjective, I think I will need to use IVT but am not sure how.
4.If f is differentiable at c then exists and is equal to f '(c).
I think this is true but don't know how to prove it, any ideas?