  A relaxation of the notion of invariant set, known as $k$-quasi-invariant
set, has appeared several times in the literature in relation to group
dynamics. The results obtained in this context depend on the fact that the
dynamic is generated by a group. In our work, we consider the notions of
invariant and 1-internally-quasi-invariant sets as applied to an action of a
function $f$ on a set $I$. We answer several questions of the following type,
where $k \in \{0,1\}$: what are the functions $f$ for which every finite subset
of $I$ is internally-$k$-quasi-invariant? More restrictively, if $I =
\mathbb{N}$, what are the functions $f$ for which every finite interval of $I$
is internally-$k$-quasi-invariant? Last, what are the functions $f$ for which
every finite subset of $I$ admits a finite internally-$k$-quasi-invariant
superset? This parallels a similar investigation undertaken by C. E. Praeger in
the context of group actions.
