# Description of Floor Function Correct?

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 – J. W. Tanner Oct 2 '20 at 12:58
• Agree with @J.W.Tanner, $\inf{x}=x$ if $x\in\mathbb{R}$, which makes your statement odd. – Weierstraß Ramirez Oct 2 '20 at 13:01
• well but $x$ is not in $\mathbb{Z}$ – user2550228 Oct 2 '20 at 13:04
• Doesn't the floor function map a real number $x$ to the largest whole number less than or equal to $x$? How about $f:x\mapsto \sup \{z\in\mathbb Z|z\le x\}$? – J. W. Tanner Oct 2 '20 at 13:06
• @J.W.Tanner you are correct. Want to post it as answer, then I can upvote. at All: thank you for your comments, i really appreciate it. – user2550228 Oct 2 '20 at 13:14

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$$(x)=\sup\{z\in\mathbb Z|z\le x\}$$.