Question

The number of integer(s) in the range of $\sqrt{\left[\text{sgn}\right(x){]}^2-\{\text{sgn}\left(x\right){\}}^2}$ is
(Here, $[.],\{.\}$ and $\text{sgn}\left(x\right)$ represent the greatest integer function, fractional part function and signum function respectively)

Answer 1

$y=\sqrt{\left[\text{sgn}\right(x){]}^2-\{\text{sgn}\left(x\right){\}}^2}$
$\text{sgn}\left(x\right)=\{\begin{array}{cc}-1,& x<0\\ 0,& x=0\\ 1,& x>0\end{array}$
$\Rightarrow \left[\text{sgn}\right(x\left)\right]=\{\begin{array}{cc}-1,& x<0\\ 0,& x=0\\ 1,& x>0\end{array}$
$\Rightarrow \left[\text{sgn}\right(x){]}^2=\{\begin{array}{cc}1,& x\ne 0\\ 0,& x=0\end{array}$

$\left\{\text{sgn}\right(x\left)\right\}=0\forall x\in R$

$\therefore y=\sqrt{\left[\text{sgn}\right(x){]}^2}$
$\Rightarrow y\in \{0,1\}$