Consider the piecewise function $R(x)=\{[x,if0x1$ and $xis$ rational $],[0,if0x1$ and $xis$ irrational $]\}$

$($a$)(4$ pt$)$ Considering any partition, $P,$ of $0,1,$ show $L(R,P)=0,$ then conclude why ${\displaystyle \underset{0}{\overset{1}{}}}{?}_R(x)dx=0.$

$($b$)(3pt)$ Consider any partition, $P,$ of $0,1$ show why $\frac{?}{\text{b}\text{a}\text{r}}({\displaystyle \underset{0}{\overset{1}{}}})R(x)dx=\frac{1}{2}.($A sketch of the graph and seeing the shape may help$)$

$($c$)(3pt)$ Is $R(x)$ integrable or not? Explain.