Composition of discontinuous functions Let $f(x) = [x]$  and $$ g(x)=\begin{cases} 0&\text{if}\;x \in \Bbb Z\\x^2&\text{otherwise}\end{cases}$$
 Is $g\circ f$ continuous?
I know conditions of continuity but in case of composition of discontinuous functions shouldn't  the composition be always discontinuous since the domain of the outer function is always  discontinuous because of the discontinuous function inside?
P.S: $[ . ]$ is greatest integer function .
 My book says it is continuous for all x.

A simple example of the composition of two discontinuous functions giving a continuous one.
$$f(x)=\{x\}$$
and
$$g(x)=x(1-x)+\lfloor x\rfloor.$$
