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?
