Column-I		Column-II
A)	$f_1\left(x\right)=\left|\right|x-6|-|x-8\left|\right|-|x^2-4|+3x-|x-7{|}^3,x\in R$	P)	$f_2\left(x\right)$ is continuous $\forall \text{x}\in \text{R}$ but not differentiable at 2 points on R
B)	$\begin{array}{l}{f}_2\left(x\right)=(x^2-9)|x^2+11x+24|+\sin |x-7|\\ +\cos |x-4|+{(x-1)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}\sin (x-1),x\in R\end{array}$	Q)	$f_1\left(x\right)$  is continuous $\forall \text{x}\in \text{R}$ but not differentiable at 5 points on R
C)	$f_3\left(x\right)=\left\{\begin{array}{ll}{(x+1)}^{3/5}-\frac{3\pi }{2}& ,x<-1\\ \left(x-\frac{1}{2}\right){\cos }^{-1}\left(4x^3-3x\right)& ,-1\le x\le 1\\ {(x-1)}^{5/3}& ,1<x<2\end{array}\right.$	R)	$f_4\left(x\right)$  is discontinuous at 3 points as well as not differentiable at 3 point on $[-1,2\pi ]$
D)	$f_4\left(x\right)=\left\{\sin x\right\}\left\{\cos x\right\}+\left({\sin }^3\pi \left\{x\right\}\right)\left(\right[x\left]\right),x\in [-1,2\pi ]$ (where [.] denotes the greatest integer function and {.} fractional part function)	S)	$f_3\left(x\right)$ is continuous  $\forall \text{x}\in \left(-\infty ,2\right)$ but not differentiable at 3 points $(-\infty ,2)$

A)  $f\left(x\right)=\left|\right|x-6|-|x-8\left|\right|-|x^2-4|+3x-|x-7{|}^3$  is continuous  $\forall x\in R$ and not differentiable at 
$x=-2,2,6,7 \& 8$
B)  $f\left(x\right)=(x^2-9)|x^2+11x+24|+\sin |x-7|+\cos |x-4|+{(x-1)}^{3/5}\sin (x-1)$  is continuous  $\forall \text{x}\in \text{R}$  and not differentiable at  $\text{x}=-8\&7$
C)  $f\left(x\right)=\{\begin{array}{l}{(x+1)}^{3/5}-\frac{3\pi }{2}:x<-1\\ (x-\frac{1}{2}){\cos }^{-1}(4x^3-3x):-1\le x\le 1\\ {(x-1)}^{5/3}:1<x<2\end{array}$ is continuous not differentiable  at  $x=-1, -\frac{1}{2}\text{ \&}1$     

D)  $f\left(x\right)=\left\{\sin x\right\}\left\{\cos x\right\}+\left({\sin }^3\pi \left\{x\right\}\right)\left(\right[x\left]\right),x\in [-1,2\pi ]$

Let,  $g\left(x\right)=\underset{\text{cont. at }x-1}{\underbrace{\left(\sin \pi \right\{x\left\}\right)\left(\right[x\left]\right)}}\left({\sin }^2\pi \left\{x\right\}\right)$

${\text{g}}^{\text{'}}\left({\text{I}}^{+}\right)={\text{g}}^{\text{'}}\left({\text{I}}^{-}\right)$  so differentiable at   $\text{x}=\text{I}$ and for  $\left\{\sin \text{x}\right\}\left\{\cos \text{x}\right\}$
Doubtful points for non differentiability are  x=0, $\frac{\pi }{2},\pi ,\frac{3\pi }{2}$ 
$\because \left\{\sin x\right\}-\left\{\cos x\right\}$  is discontinuous at  x=0,  $\frac{\pi }{2},2\pi$
 So not differentiable at  $x=2n\pi ,2n\pi +\frac{\pi }{2}$