Is my description of the floor function correct?
$$ f = \begin{cases} \mathbb{R} \rightarrow \mathbb{Z} \\ x \mapsto z = \inf(x) \end{cases} $$
Explanation:
The floor function maps a real number $x$ to the smallest whole number less than or equal to $x$. The infimum of is the largest lower bound of a set. The above stated function $f$ maps a real number $x$ to the largest whole number $z$ for which $z \leq x$, which is the definition of the floor function. Hence $f = \operatorname{floor}$.
Inf should be defined on a set, not one real number.
Furthermore, the floor function maps a real number $x$
to the largest integer less than or equal to $x$,
so it could be defined as floor$(x)=\sup\{z\in\mathbb Z|z\le x\}$.