# Using a lattice to guarantee niceness of a function

Let's say that $\Lambda_\varepsilon$ is a lattice over $\mathbb R^m$ generated by the unit vectors scaled by $\varepsilon$ and let $f:\mathbb R^n\to\mathbb R^m$ be a continuous function. $\Lambda_\varepsilon$ gives us a natural way to specify niceness constraints on $f$. For example, we could say

A function, $f$, is somewhat nice if there exists an $\ell$ such that for all $\ell> \varepsilon >0$, $f$ passes through every cell of the lattice $\Lambda_\varepsilon$ finitely many times.

or we could say

A function, $f$, is $k$-nice if there exists an $\ell$ such that for all $\ell>\varepsilon>0$, there exists a $k$ such that $f$ passes through every cell of the lattice $\Lambda_\varepsilon$ at most k many times.

or we would say

A function, $f$, is very nice if there exists an $\epsilon$ such that for all $\ell>\varepsilon>0$, $f$ passes through every cell of the lattice $\Lambda_\varepsilon$ at most once.

These conditions are pretty weak, but they do impose a rather natural notion of niceness on $f$. Do these conditions have a name? Are they strong enough to prove a common niceness condition (perhaps when combined with something else)? If these definitions haven't been studied, are there other conditions defined by regulating the behavior of a function in lattice cells that have been studied?

The idea for this came out of trying to regulate the behavior of a function to preclude space-filling curves. Space-filling curves are very squirrely, and will frequently enter and reenter the same cell. It's easy to see that a number of conditions are sufficient to satisfy these niceness conditions, but I haven't found any that are equivalent or that are stronger. I know that continuity is too weak, and that piece-wise linearity is much stronger, but I haven't had much luck identifying anything that seems to be "approximately as strong" as these conditions.