Consider functions $f:\mathbb{R}\rightarrow \mathbb{R}$ with the property that $\left | f\left ( x \right )-f\left ( y \right ) \right |\leq 4321\left | x-y \right |$ for all real numbers $x,y$. Then which one of the following statement is true?
- $f$ is always differentiable
- There exists at least one such $f$ that is continuous and such that $\underset{x\rightarrow\pm \infty }{lim}\:\frac{f\left ( x \right )}{\left | x \right |}=\infty$
- There exists at least one such $f$ that is continuous, but is non-differentiable at exactly $2018$ points, and satisfies $\underset{x\rightarrow\pm\infty }{lim}\:\frac{f\left ( x \right )}{\left | x \right |}=2018$
- It is not possible to find a sequence $\left \{ x_{n} \right \}$ of real numbers such that $\underset{n\rightarrow\infty }{lim}\:x_{n}=\infty$ and further satisfying $\underset{n\rightarrow\infty }{lim}\left | \frac{f\left ( x_{n} \right )}{x_{n}} \right |\leq 10000$
