We say that a function $f: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$ is *uniformly continuous* if there is an integer $K\geq 1$ such that
 whenever $(x,y),(x',y')\in \mathbb{Z}\times \mathbb{Z}$ with $|(x,y)-(x',y')| = 1$ in the Euclidean distance, then $|f(x,y)-f(x',y')| < K$.


Is there an injective uniformly continuous function $f:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$?



Injective uniformly continuous function $f:\mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}$?
No. A uniformly continuous function takes $O(N)$ distinct values on an $N\times N$ grid.


