I consider the integral $\int_0^1 f$.
I was asking myself a question about the Riemann integral. If I take regular subdivision $\sigma_n$ $(0<1/n<2/n<\ldots<(n-1)/n<1)$ smaller and smaller, and I find that $\overline{S}_{\sigma_n}-\underline{S}_{\sigma_n}\ge c>0$, can I conclude that the function is not integrable? It seems to me I cannot conclude this because every subdivision cannot be refined in a regular subdivision (is this true). It seemed to me that for example the subdivision $0<1/\sqrt{2}<1$ cannot be refined in a regular subdivision. If I cannot conclude indeed, would someone have an example. If we could conclude, why is that?
Here $\overline{S}_{\sigma_n}$ refers to the upper Darboux sum with respect to the subdivision $\sigma_n$ and $\underline{S}_{\sigma_n}$ refers to the lower Darboux sum with respect to the subdivision $\sigma_n$