Given that $$\ f(x)=\begin{cases} 0, & \text{if $x$ is rational} \\ 1, & \text{if $x$ is irrational}\end{cases}$$
I want to show that the integral $$\int_0^1 f(x)\,dx$$ does not exist by using a Riemann sum and showing that it does not converge to a number.
I'm guessing that the Riemann sum that I'm searching for should look like this: $$a_n = \sum_{i=1}^n \left(f\left(\frac{i-c}{n}\right)*\frac{1}{n}\right),\ c\in [0,1]$$
and that I'm supposed to show that $$\lim_{n \to \infty} a_n$$ does not exist.
Edit: I knew that the Dirichlet function is not continuous on [0,1] and that therefore it is not integrable when I made this post. This is the reasoning that I've seen for answers to similar questions.
I was wondering if it was possible to demonstrate that it is not integrable by coming up with a Riemann sum that does not converge.
Additionally, in the form that I suggested above, the variable c does not have to be rational.