Given,

$f\left(x\right)=\left\{\begin{array}{l}x+1, x<0\\ \left|x-1\right|, x\ge 0\end{array}\right.$and $g\left(x\right)=\left\{\begin{array}{l}x+1, x<0\\ 1, x\ge 0\end{array}\right.$

$\Rightarrow f\left(x\right)=\left\{\begin{array}{lc}x+1,& x<0\\ 1-x,& 0\le x<1\\ x-1,& x>1\end{array}\& g\left(x\right)=\left\{\begin{array}{cc}1+x,& x<0\\ 1,& x\ge 0\end{array}\right.\right.$
     

Now, for $gof\left(x\right)$ 

$gof\left(x\right)=\left\{\begin{array}{cc}1+f\left(x\right),& f\left(x\right)<0, x<0\\ 1,& f\left(x\right)\ge 0, x\ge 0\end{array}\right.$

$\Rightarrow gof\left(x\right)=\left\{\begin{array}{cc}x+2,& x<-1\\ 1,& x\ge -1\end{array}\right.$

Now plotting the diagram we get,

Now from above diagram we can say that function is continuous but not differentiable at $x=-1$ due to sharp corner.