Let $f : \mathbb{R} \to \mathbb{ R}$ be such that $f' (x)$ exists for all non zero $x$ and $\lim\_{x\to 0} f' (x) = 0$.

Then

(i) $f$ is continuous but not differentiable at $0$.

(ii) $f$ is differentiable at $0$ and $f' (0) = 0.$

(iii) $f$ has either a local maximum or a local minimum at $0$.

(iv) None of the above.

I am totally clueless. Thank you for help and discussion.