What do you call this generalized floor function?

Question

Given

1. an $n$-tuple of ordinal values $\mathbf{x}$, the members of which are each comparable to one another and

2. a single value $y$, $\mathrm{min}(\mathbf{x})\le y$, which is comparable to any of the members of $\mathbf{x}$,

maximize $x\in \mathbf{x}$, $x\le y$.

(This seems to be a simple, special case among "constrained extrema" problems.)

In terms of the range, this function is like the floor function, except that the range is constrained to $\mathbf{x}$ rather than to the integers.

For example, suppose $\mathbf{x}=(-100, -10, 0, 10, 100)$ and $y=-11$. Then $x=-100$. And if $y=-10$, $x=-10$.

Provided that the constraints are met, I believe this can be expressed in R as max(x[x <= y]).

One application is to the selection of aesthetic boundaries for a plot.