Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function. For an $\alpha \in \mathbb{R}$, suppose that
$\lim _{h \rightarrow 0} \frac{f(\alpha+h)-f(\alpha-h)}{h}=p,$
where $p \in \mathbb{R}$. Then

• A

  $f$ is differentiable at $\alpha$ if the left hand and right hand derivatives at $\alpha$ exist
• B

  $f$ is differentiable at $\alpha$ and $f^{\prime}(\alpha)=p$
• C

  $f$ is differentiable at $\alpha$ and $f^{\prime}(\alpha)=\frac{p}{2}$
• Correct answer
  D

  $f$ may not be differentiable at $\alpha$

Hence, the correct answer is option: 'D'.