Match the following Column - I (functions) and Column - II (properties)

$\text{(A)}f\left(x\right)=x\left|x\right|=\{\begin{array}{cc}-x^2,& x<0\\ x^2,& x>0\end{array}$

is continuous and differentiable everywhere also increasing.

$\text{(B)}f\left(x\right)=\sqrt{\left|x\right|}=\{\begin{array}{ll}\sqrt{-x},& x<0\\ \sqrt{x},& x\ge 0\end{array}$

$\Rightarrow f^{\mathrm{\prime }}\left(x\right)=\{\begin{array}{cc}-\frac{1}{2\sqrt{-x}},& x<0\\ \frac{1}{2\sqrt{x}},& x>0\end{array}$

Continuous everywhere differentiable everywhere except at $x = 0$. Not increasing.
(C) $f\left(x\right) = x + \left[x\right]$At integral point $x = I$,
$LHL = I + (I – 1) = 2I – 1, RHL = I + I = 2I = f\left(I\right),$
So not continuous hence not differentiable at integral points but increasing.

$\text{(D)}f\left(x\right)=|x-1|+|x+1|=\{\begin{array}{ccc}-2x& ,& x<-1\\ 2& ,& -1\le x<1\\ 2x& ,& 1\le x\end{array}$

Continuous everywhere, differentiable everywhere but not increasing in (1,1).