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

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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 $X$.

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    $\begingroup$ I got confused for a moment, until I saw that Hans's function have domain the natural numbers! So $|y-x|\geq 1$ is just $x\not=y$. So, yeah, Lipschitz it is. $\endgroup$ – Matthew Daws Jul 8 '10 at 19:45

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