Suppose I have a function that is not quasiconvex, as in the graph below, but would be quasiconvex if we cared only about integer points. That is,
$f:X \subset \mathbb{Z}\rightarrow \mathbb{R}$
and for any $x,y\in X$ and $u\in X$ such that $x<u<z$, then $f(z) \leq \max\{f(x),f(y)\}$.
Would I refer to this function as quasiconvex on the integers? Wikipedia's definition of quasiconvexity admits only functions with convex domains, but this extension seems quite natural. Would there be another concept to which I could appeal?
