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