Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying $|f(x)-f(y)| \leq|x-y|^{\frac{3}{2}} \quad$ for all $x, y \in \mathbb{R}$.

• A

  $|f(x)| \leq|x|^{\frac{3}{2}}$ for all $x \in \mathbb{R}$.
• Correct answer
  B

  there exists $M \in \mathbb{R}$ such that $|f(x)| \leq M$ for all $x \in \mathbb{R}$.
• Correct answer
  C

  $f$ is differentiable everywhere.
• D

  $f(1) \lt f(0)$

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