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


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 $X$.

| cite | improve this answer | |
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.