Does there exist a bounded set $A \subset \mathbb{R}^ n$ (for some $n$) and a function $f:A\to A$ (not necessarily onto) such that $\forall x,y \in A$ then $\|x-y\| < \|f(x)-f(y)\|$?

If such a set $A$ (and a function $f$) exists, does there even exist such a set $A$ and a function $f$ such that $f$ is continuous? (Or - can it be proved that such a continuous $f$ doesn't exist?)