Another definition of Riemann integrability

Question

This is the definition of Riemann integrability in terms of Riemann sums instead of upper and lower Darboux sum definition. I want to show using ONLY this definition that The Dirichlet function $f : [0, 1] → \mathbb R$, defined by $$f(x) = \begin{cases} 1, & x ∈ \mathbb Q \\ 0, & x ∈ [0, 1] - \mathbb Q \end{cases}$$ is not Riemann Integrable. How do we do that? How do we negate the statement? Can anyone help?

Answer 1

I believe this is the original definition of the Riemann integral. To show that the Dirichlet function is not Riemann integrable according to this definition, we have to show that $\forall B\in\mathbb{R}, \exists \epsilon >0 : \forall \delta >0, \exists P\in\mathcal{P} : |P|<\delta$ and there exists a Riemann sum $S(P,f)$ such that $|S(P,f)-B|\ge \epsilon$ where $\mathcal{P}$ is the set of all partitions of $[0, 1]$.

For simplicity, let us just consider the case where $B\in [0, 1]$ since $0\le f\le 1$. Let $\epsilon=1/2$. Let $\delta>0$ be arbitrary, $n\in\mathbb{N}$ be large enough such that $1/n<\delta$ and let $P=(0, 1/n, 2/n, \dots, (n-1)/n, 1)$. By construction, $|P|=1/n<\delta$ and $S(P,f)\in \{0, 1/n, 2/n, \dots, (n-1)/n, 1\}$ for any choice of Riemann sum $S(P,f)$ because the intermediate points can either be rational or irrational. Now, for some $k=0, 1,\dots, n$, $|B-k/n|\ge 1/2$ and so on $[(j-1)/n, j/n]$, we can pick the intermediate point $\xi_j$ to be rational if $j\le k$ and irrational if $j>k$. This way, $S(P,f)=k/n$ and $$|S(P,f)-B|=|B-k/n|\ge 1/2$$