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- Mar 10, 2012

- 835

I am sorry that I haven't been able to take part in discussions lately because I have been really busy.

I am having trouble with a question.

In a past year paper of an exam I am preparing for it read:

Let $f: (a,b)\to \mathbb R$ be a differentiable function with $f'(x)\neq 0$ for all $x\in(a,b)$. Then is $f$ necessarily injective?

I know that a function can be differentiable at all points and have a discontinuous derivative.

This makes me think that $f$ is not necessarily injective. But I am not able to construct a counterexample.

Can anybody help?