BALKAN-SL 2013 A7

Suppose that $k$ is a positive integer. A bijective map $f : Z \to Z$ is said to be $k$ -jumpy if $|f(z) - z| \le k$ for all integers $z$ . Is it that case that for every $k$ , each $k$ -jumpy map is a composition of $1$ -jumpy maps? It is well known that this is the case when the support of the map is finite.