Let $(X,d_x),(Y,d_Y)$ be bounded metric space. Let $f:X\rightarrow Y$ be a homeomorphism. Is it true that there exist $a,b>0$ such that $$ad_X(x_1,x_2)<d_Y(f(x_1),f(x_2))<bd_X(x_1,x_2)$$ for any $x_1,x_2\in X$?
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Think of the map $y=x^2$ of the unit interval. – Moishe Kohan Nov 11 '13 at 17:00
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The maps with the property you described are called bi-Lipschitz homeomorphisms. This is a strictly stronger property than being a homeomorphism. Two concrete examples are $f(x)=x^2$ on $[0,1]$ and its inverse (taken from comment by studiosus). A more abstract example: let $(X,d)$ be any metric space, and consider the identity map from $(X,d)$ to $(X,d^{1/2})$. Here $d^{1/2}$ means the metric $\sqrt{d(x,y)}$ on the same set. Unless $X$ is a discrete space, $f$ is not Lipschitz.
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