In need for something equivalent to the continuity-definition of real functions I use the following definition of "coarse-continuity" for sequences. Has it been known already? Has it even got a name? 


Definition: A function $f(x)$ with $x \in \mathbb{N}$ is called coarsely continuous if and only if there exists a fixed positive constant $C$ such that  

${\forall}$ $x, y \in \mathbb{N}$, $|y-x| \ge 1$ : $\dfrac{|f(y) – f(x)|}{|y-x|} < C$. 



Has coarse continuity been known already?
Functions like this are called [Lipschitz](http://en.wikipedia.org/wiki/Lipschitz_continuity). 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 $X$.


