The Hölder condition with the negative exponent $|f(x) - f(y)| \leq \frac{1}{|x-y|}$

Question

Suppose $f: \mathbb{R} \to \mathbb{R}$ is such that $\forall \ x, y: x \neq y \to$ $|f(x) - f(y)| \leq \frac{1}{|x-y|}$

I need to prove that $f$ is

A) bounded

B) continuous

C) differentiable

D) non decreasing

on $\mathbb{R}$

Could you please give me any hints?

Thanks a lot in advance!

Answer 1

Fix a point $a \in \mathbb R$. The stated hypothesis tells you that $$ \lim_{x \to \infty} f(x) = f(a).$$

Can you take it from there?