$ U:=\{f:\mathbb{R}\to\mathbb{R} : f(\mathbb{R}) \subset [0,1]\}$
$ V:=\{f:\mathbb{R}\to\mathbb{R} : f(\mathbb{Z}) =\{0\}\ \}$
I am a little confused by the notation $f(\mathbb{R})$ and $f(\mathbb{Z})$ and to be sure I'd like confirmation/correction. I am assuming that it means that $U$ is the set of functions for which the output values are contained in $[0,1]$ for all real input, such as $f:\mathbb{R}\to\mathbb{R}, x\mapsto \frac{1\ +\ \sin x}{2}$ or $f\equiv1 $. $V$ is the set of functions for which output is $0$ for all integer input, such as $f:\mathbb{R}\to\mathbb{R}, x\mapsto \sin \pi x$ or $f\equiv 0$.
Yes you are completely right.
More generally if $f \colon X \to Y$ is any map, then for $Z \subseteq X$ we write in some abuse of notation $f(Z)=\{f(z) \mid z \in Z\}$