 Solution

Not injective; its image of a function is exactly $\{1,2,3,\dots \}$. For example, both $x=e^{-1}$ and $x=e^{-1/2}$ give the ceiling function value $1$. Since $0<x<1$, $(\ln x{)}^2>0$, so the value $0$ is never attained. For every positive integer $j$,Each such interval is nonempty and contained in $(0,1)$, with the upper endpoint $1$ excluded when $j=1$. Thus the answer concerning surjectivity depends on the convention for the natural numbers: onto if $\mathbb{N}=\{1,2,\dots \}$, and not onto if $0\in \mathbb{N}$. The source does not settle that convention.