Let $g\colon[0,1]\rightarrow \mathbb{R}$, be defined as $g(x) = 0$ if $x \in \mathbb{Q}$ and $g(x)=1/x$ if $x \not\in\mathbb{Q}$. Show that there exists a sequence $(P_n)$ of partitions of $[0,1]$ such that $||P_n||\to0$ and$\lim\limits_{n\to\infty} S(g,P_n)$ exists.
Why is $g\not\in\mathscr{R}[0,1]$?
I have tried defining a number of sequences with a $1/n$ or $1/n^2$ factor of distance between the intervals of each partition, but this doesn't seem to work, as I could just pick a $t_{1} \in [x_0, x_1]$ tag close enough to zero, such that the $f(t_k)$ factor of the Riemann sum would blow up for each partition, making the limit nonexistent.
I guess what I'm not understanding correctly is the way the $t_{k}$ tags for the sums are chosen. Is the sequence of partitions supposed to converge for any choice of $t_{i}$ in every subinterval? Or should I select a previously fixed set of $t_{i}$ tags for every partition such that the sequence converges? Because in that case, this question would be trivial, and I could just select the tags as rational numbers.
I think what you need to construct such a sequence is that any open interval , no matter how small it is, contains a rational and an irrational. Using this, you just pick a rational in every subinterval and the Riemann sum will always be 0 (the limit will also be 0).
Similarly, if you choose an irrational for every subinterval, the limit goes to infinity. This means that g is not Riemann integrable on [0,1].
The Riemann integrability is, by definition, as long as ||P|| is small enough, no matter what tags you choose, the sum can be arbitrarily close to certain number (integral). So if you have a sequence of partitions and fixed sample points whose sum converges, it does not implies the integrability. Even if you have a sequence of partitions such that for arbitrary choice of tags, the sums converge to a number, you can not derive the integrability because there might be another sequence of partitions and choices of tags whose sums do not converge to that number.
However, if you have already known the integrability, you can easily compute the value of an integral by choosing any sequence of partitions, choosing any tags and calculating the limit of the sums.