If $f\left(x\right)=\left\{\begin{array}{cl}\frac{x\left(3e^{{\displaystyle \frac{1}{x}}}+4\right)}{2-e^{{\displaystyle \frac{1}{x}}}},& x\ne 0\\ 0,& x=0\end{array}\right.$, then $f\left(x\right)$ is

$$ Given $f\left(x\right)=\left\{\begin{array}{ccc}{\log }_a{\left(a\left|\left[x\right]+[-x]\right|\right)}^x\left(\frac{a^{{\displaystyle \frac{2}{\left(\frac{\left[x\right]+\left[-x\right]}{\left|x\right|}\right)}-5}}}{3+a^{\frac{1}{\left|x\right|}}}\right)& \text{for}& x\ne 0\\ 0& \text{for}& x=0\end{array}\right.$; $a>1$ where [.] represents the integral part function, then

If $f\left(x\right)=\left\{\begin{array}{ll}\frac{x^2-1}{x^2+1}& , 0<x\le 2\\ \frac{1}{4}\left(x^3-x^2\right)& , 2<x\le 3\\ \frac{9}{4}\left(\left|x-4\right|+\left|2-x\right|\right)& , 3<x<4\end{array}\right., \text{then}$

Let $f\left(x\right)$ be defined in$[-2,2]$ by $f\left(x\right)=\left\{\begin{array}{ll}\max (\sqrt{4-x^2},& \sqrt{1+x^2}), -2\le x\le 0\\ \min (\sqrt{4-x^2},& \sqrt{1+x^2}), 0<x\le 2\end{array}\right.$, then $f\left(x\right)$

Consider the following statements:
${\mathrm{S}}_1:$ Number of points where $f\left(x\right)=\left|xsgn\left(1-x^2\right)\right|$ is non-differentiable is $3$
$S_2:$ Defined $f\left(x\right)=\left\{\begin{array}{ll}a\sin \left(\frac{\pi }{2}\left(x+1\right)\right)& , x\le 0\\ \frac{\tan x-\sin x}{x^3}& , x>0\end{array}\right.$, in order that, $f\left(x\right)$ is continuous,$\text{'}a\text{'}$should be equal to $\frac{1}{2}$

${\mathrm{S}}_3:$ The set of all points, where the function $\sqrt[3]{x^2\left|x\right|}$ is differentiable is $(-\infty ,0)\cup (0,\infty )$
${\mathrm{S}}_4:$ Number of points where $f\left(x\right)=\frac{1}{{\sin }^{-1}\left(\sin x\right)}$ is non-differentiable in the interval $(0,3\pi )$ is $3 .$ State, in order, whether $S_1,S_2,S_3,S_4$ are true or false

Consider the following statements:

${\mathrm{S}}_1:$ Let $f\left(x\right)=\frac{\sin \left(\pi \left[\mathrm{x}-\pi \right]\right)}{1+[\mathrm{x}{]}^2}$ where $[.]$ stands for the greatest integer function. Then $f\left(x\right)$ is discontinuous at $x=n+\pi , n\in I$.
${\mathrm{S}}_2:$ The function $f\left(x\right)=p[x+1]+q[x-1]$, where $[.]$ denotes the greatest integer function is continuous at $x=1$ if $p+q=0$.
${\mathrm{S}}_3:$ Let $f\left(x\right)=\left|\right[x] x|$ for $-1\le x\le 2,$ where $[.]$ is greatest integer function, then $f$ is not differentiable at $x=2$.
${\mathrm{S}}_4:$ If $f\left(x\right)$ takes only rational values for all real $x$ and is continuous, then $f^{\text{'}}\left(10\right)=10$.

Mark $\mathrm{F}$ if the statement is false and $\mathrm{T}$ if the statement is true.