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I am working with a function $f : \mathbb{R}^N \to \mathbb{R}$ having the property that for every $R > 0$, there exists $M > 0$ such that if $x, y \in \mathbb{R}^N$ and $\vert x - y \vert \le R$, then $$ \vert f (x) - f (y)\vert \le M. $$ By the triangle inequality (and since $\mathbb{R}^N$ is a convex set), it is sufficient for this property to hold for some $R > 0$.

These are functions that have a finite “modulus of continuity”, provided you forget about the classical continuity condition on a modulus of continuity. It can be interpreted as a function of bounded maximal oscillation at every scale. Sufficient conditions for this condition are boundedness or uniform continuity.

Equivalently, these functions are characterized by the condition that for every $R > 0$ or for some $R > 0$, $$ \sup_{a \in \mathbb{R}^N} \sup_{x, y \in B_R (a)} \vert f (x) - f (y)\vert < \infty. $$

My question is whether such functions have a name at some place in the mathematical literature. Could such function appear naturally in the analysis of mappings between metric spaces?

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The term is coarsely continuous, also known as asymptotically continuous or large-scale continuous.

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  • $\begingroup$ Thanks to your answer, I have had a look to John Roe’s Coarse geometry, and he would use the word bornologous, and in this case, it is equivalent to large-scale continuity. $\endgroup$ Jun 6, 2016 at 14:56
  • $\begingroup$ Yes, I see that that term is used also. I guess the terminology has not stabilized. The terms I suggested for your property are given in Recent Progress in General Topology III, p. 168 (that page isn't available on Google books but you can get the entire book at SciHub). $\endgroup$
    – Nik Weaver
    Jun 6, 2016 at 15:32

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