Chapter 5.5, Problem 43ES

It can be shown that every interval contains both rational and irrational numbers. Accepting this to be so, do you believe that the function

$f\left(x\right)=\left\{\begin{array}{lll}1\hfill & \text{if}\hfill & x\text{ is rational}\hfill \\ 0\hfill & \text{if}\hfill & x\text{ is irrational}\hfill \end{array}\right.$

is integrable on a closed interval $\left[a.b\right]$ ? Explain your reasoning.