ca.analysis and odes - Has coarse continuity been known already?

Functions like this are called Lipschitz. The definition works for maps between any two metric spaces. There is also the notion of being coarse lipschitz:

If you have a function $f : X to Y$ between two metric spaces, and constants $K geq 1$ and $C geq 0$, then $f$ is $(K,C)$--coarse lipschitz if
$d_Y(f(x),f(y)) leq K d_X(x,y) + C$ for any $x$ and $y$ in $X$.