Suppose that $(X,d_X)$ and $(Y,d_Y)$ are complete doubling metric spaces and let $f:X\rightarrow Y$ be a non-constant Lipschitz map. Then can does there exist a lsc function $\rho:(0,\infty)\rightarrow (0,\infty)$ with the properties that

- $ \lim\limits_{t \uparrow \infty}\rho(t) = \infty $
- $ \lim\limits_{t \downarrow 0}\rho(t) = 0, $

Such that for every $x,y \in X$, $$ \rho(d_X(x,y)) \leq d_Y(f(x),f(y)) \leq Lip(f) d_X(x,y) , $$ where $Lip(f)$ is the best Lipschitz constant of $f$.