Let  $f:R\to R$ be a function by  $f\left(x\right)=\min \left\{x+1,\left|x\right|+1\right\}$. Then which of the following true ? 

$x=0\Rightarrow \left|x\right|=x$

$f\left(x\right)=\min \left\{x+1,x+1\right\}=x+1$

$x<0,\left|x\right|=-x$

$f\left(x\right)=\min \left\{x+1,-x+1\right\}$

$-x+1>x+1$

$\therefore x<0,f\left(x\right)=x+1,x>0,f\left(x\right)=x+1$

$\therefore f\left(x\right)=x+1\forall x$    $\left(\begin{array}{l}\because y=x+1,\\ f\text{'}\left(x\right)=1\end{array}\right)$

 $f\left(x\right)$ is differentiable every where (continuous and constant slope)