Prove that the range of $f$ is all of $\mathbb{R}$. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continous function such that  $|f (x)−f (y)| \geqslant |x −y|$, for all real $x$ and $y$. I need to prove that the range of $f$ is all of $\mathbb{R}$. I tried to solve the inequality to get a general form for the functions, but it just ended up in a mess. 

If $f$ is injective and continuous it must be strictly monotonic. By looking at $-f$ if necessary, we can assume that $f$ is strictly increasing and so we have $f(x)-f(y) \ge x-y$ whenever $x \ge y$.
In particular, $\lim_{x \to \infty} f(x)-f(y) = \infty$ and
$\lim_{y \to -\infty} f(x)-f(y) = \infty$.
Hence the range is unbounded in both directions and the intermediate function theorem shows that the range must be $\mathbb{R}$.
