Suppose $a$ is a real number and $f$ is defined as
$f(x)= \begin{cases}|x|^a \sin \left(|x|^{-3}\right) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{cases}$
Then

• A

  $\approx f$ is continuous everywhere.
• Correct answer
  B

  $f^{\prime}(0)$ exists if and only if $a \gt 1$.
• C

  if $a=4$, then $f^{\prime}$ is continuous at 0
• Correct answer
  D

  $f^{\prime}$ exists and is continuous on $\mathbb{R}$ if and only if $a \gt 4$.

Hence, the correct answer is options: 'B', 'D'.