real analysis - Can this function be integrated and, if so, what is its value?

Say I have a piecewise function defined as:

$f(x) =
\begin{cases}
0 & \text{if x is irrational} \\
1/q &\text{if x = $p/q$, where $p,q > 0$ are relatively prime integers} \\
\end{cases}$

over the interval $[0,1]$. I can prove that $f$ is continuous over the irrational numbers and discontinuous at all rational numbers. But, is the function Riemann integrable:

$\int_{0}^{1}f(x)dx$? If so how would I prove it and what would be the value? Intuition suggests that the value would be zero but I'm at a loss for how to proceed. Assuming the function IS Riemann integrable could I split it up into multiple sections at every discontinuity? But, there are infinitely many discontinuities...

Thanks for your help.

Answer

By Lebesgue's Criterion, a function is Reimann integrable iff it is bounded and the set of discontinuities has measure $0$, i.e. countable. In this case, $|f|\leq1$, and the set of discontinuities is $\mathbb{Q}$ which is countable and thus measure $0$. So, the function $f$ is Reimann integrable.

For any function which is Reimann integrable, it is also Lebesgue integrable, and the value of each integral is equivalent. Hence, it suffices to find the Lebesgue integral of $f$. To do this, consider that $f$ is piecewise constant, and nonzero on a set of measure $0$, so it has Lebesgue integral $\int f = 0$, and so the Reimann integral is also $0$.