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