Let $f:\mathbb N\to\mathbb N$ be a function such that $|f(n+1)-f(n)|\le 1$, for all $n\in\mathbb N$. In other words, the function can increase, decrease or stay constant, but the difference at each step is at most one from the previous value. The function does not "jump".
It seems to me that this is somewhat analogous to continuous real functions with bounded derivative, but I don't know enough to draw a precise connection.
Is there a common specific name for this kind of functions?