Let $\left[x\right]$ denote the greatest integer function and $f\left(x\right)=\max \{1+x+\left[x\right],2+x,x+2\left[x\right]\}, 0\le x\le 2$.  Let  $m$ be the number of points in  $[0,2]$, where  $f$  is not continuous and  $n$ be the number of points in  $(0,2)$, where   is not differentiable. Then  ${(m+n)}^2+2$ is equal to

Let,

$$\begin{gathered} g\left(x\right)=1+x+\left[x\right]=\{\begin{array}{l}1+x;x\in [0,1)\\ 2+x;x\in [1,2)\\ 5;x=2\end{array} \\ \lambda \left(x\right)=x+2\left[x\right]=\{\begin{array}{l}x;x\in [0,1)\\ x+2;x\in [1,2)\\ 6;x=2\end{array} \\ r\left(x\right)=2+x \\ f\left(x\right)=\{\begin{array}{l}2+x;x\in [0,2)\\ 6;x=2\end{array} \end{gathered}$$

$f\left(x\right)$  is discontinuous only at   $x=2\Rightarrow m=1$   

$f\left(x\right)$  is differentiable in      $(0,2)\Rightarrow n=0$