does there exist any bounded set A in $\mathbb{R}_n$ (for some n) and any function f from A to A (not necessarily onto) such that $\forall x,y \in A$ distance(x,y) < distance(f(x),f(y))? And if such set A (and function f) exists, does there even exist such set A and function f such that f is continuous? (Or - can it be proved that such continuous f doesn't exist?)