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.