TITLE: If $|f(x)-f(y)|\geq \frac12|x-y|$, must $f$ be bijective?
QUESTION [8 upvotes]: Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous function such that $$|f(x)-f(y)|\geq \frac12|x-y|$$ for all$x,y\in \mathbb R$. Then is $f$ one-one and onto?
Let $f(x)=f(y)$ i.e. $0=|f(x)-f(y)|\geq (1/2)|x-y|$ i.e $x=y$ 
Hence $f$ is injective.
But I am unable to conclude whether $f$ is onto.Any help

REPLY [7 votes]: $f$ is one to one and continue, therefore, $f$ is stricly increasing or strictly decreasing. Let us show that $f$ is not bounded, 
Let $y=0$, then $-|f(x)-f(0)|\leq \frac{x}{2}\leq |f(x)+f(0)|$
Therefore $|f(x)|\geq \frac{x}{2}-|f(0)|$ and so there is no upper bound and $|f(x)|\leq \frac{x}{2}+|f(0)|$ therefore $f$ is not low bounded, therefore $f$ is onto.