TIFR-2020-Maths-A: 5

Question

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function that satisfies:

                   $|f\left ( x \right )-f\left ( y \right )|\leq \left | x-y \right |\left | sin\left ( x-y \right ) \right |$, for all $x,y \in\mathbb{R}$.

Which of the following statements is correct?

• A. $f$ is continuous but need not be uniformly continuous.
• B. $f$ is uniformly continuous but not necessarily differentiable.
• C. $f$ is differentiable, but its derivative may not be continuous.
• D. $f$ is constant.