Let $f _1: R \rightarrow R f _2:\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow R , f _3:\left(-1, e ^{\frac{\pi}{2}}-2\right) \rightarrow R$ and $f _4: R \rightarrow R$ be functions defined by
$(i)$ $f _1(x)=\sin \left(\sqrt{1- e ^{-x^2}}\right)$
$(ii)$ $f_2(x)=\left\{\begin{array}{ll}\frac{|\sin x|}{\tan ^{-1} x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{array}\right.$, where the inverse trigonometric function of $\tan ^{-1} x$
assume values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$,
$(iii)$ $f_3(x)=\left[\sin \left(\log _c(x+2)\right)\right]$, where, for $t \in R ,[t]$ denotes the greatest integer less than or equal to $t$,
(iv) $f_4(x)=\left\{\begin{array}{ll}x^2 \sin \left(\frac{1}{x}\right) & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{array}\right.$
| $LIST I$ |
$LIST II$ |
| $P$ The function $f _1$ is |
$1$ $NOT$ continuous at $x=0$ |
| $Q$ The function $f _2$ is |
$2$ continuous at $x =0$ and $NOT$ differentiable at $x =0$ |
| $R$ The function $f_3$ is |
$3$ differentiable at $x=0$ and its derivative is $NO$T continuous at $x =0$ |
| $S$ The function $f _4$ is |
$4$ differentiable at $x =0$ and its derivative is continuous at $x =0$ |
The correct option is:
