Continuing with Yakov's comment.

We can construct a left-adjoint function \$$L : \mathbb{R}\to\mathbb{N} \$$ to our \$$I : \mathbb{N}\to\mathbb{R}\$$ such that

$$L = \begin{cases} 0 & \text{if } x\le0\\\\ \lceil x \rceil & \text{if } x>0 \end{cases}$$

Can we construct a right-adjoint function \$$R : \mathbb{R}\to\mathbb{N} \$$ to our \$$I : \mathbb{N}\to\mathbb{R}\$$ such that

$$R = \begin{cases} ? & \text{if } x\lt0\\\\ \lfloor x \rfloor & \text{if } x \ge0 \end{cases}$$

...what do we do when \$$x \lt0 \$$?