TITLE: how to prove, $f$ is onto if $f$ is continuous and satisfying $|f(x) - f(y)| ≥ |x - y|$ for all $x,y$ in $\mathbb{R}$
QUESTION [11 upvotes]: let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that
$|f(x) - f(y)| ≥ |x - y|$ for all $x,y$ in $\mathbb{R}$.
then $f$ is onto.

REPLY [3 votes]: If $M$ is connected and $f$ is continuous, then $f(M)$ is connected. $\mathbb{R}$ is connected. Hence $f(\mathbb{R})$ must be an interval. Now you only have to prove that $f(\mathbb{R})$ cannot be bounded from above or below. That's up to you.