Composition of discontinuous functions

Question

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.

Answer 1

$g\circ f$ is not continuous at $\sqrt n$ for $n \in \mathbb N$ as

$(g \circ f)(x) = n-1$ for $x \in (\sqrt{n -1}, \sqrt {n})$ and $(g \circ f)(x) = n$ for $x \in (\sqrt{n}, \sqrt {n+1})$.

And the composition of two discontinuous maps may be continuous.

Take for example

$$f(x)= \begin{cases} 0 & \mbox{for} & x\le 0\\ 1 & \mbox{for} & x>0 \end{cases}$$ and $$g(x)= \begin{cases} 0& \mbox{for} & x\le -1\\ 1 & \mbox{for} & x>-1 \end{cases}$$

$g\circ f$ is a constant map equal to $1$.

Answer 2

A simple example of the composition of two discontinuous functions giving a continuous one.

$$f(x)=\{x\}$$

and

$$g(x)=x(1-x)+\lfloor x\rfloor.$$