Some days ago as I had asked as to how to test the Riemann Integrability of the function. Now I was recently given this question about proving that the given function is Riemann integrable.
How can i show that this function is Riemann integrable or not in the interval $[0,1]$. I tried using partitions, but it didn't work. Here I don't want to use the Riemann Lebesgue lemma as iI want to understand the methodology behind selecting the partitions.
Best Answer
The function is Riemann-integrable because it is bounded (it takes values in $[0,1]$) and has countably many discontinuities, namely, the points of the form $\frac{1}{n}$ and $0$.
This uses Lebesgue's criterion for Riemann integrability which you probably meant with Riemann-Lebesgue-Lemma and hence unfortunately didn't want to use.
As for doing it by hand with partitions, try Akhil Mathew's hint above.