You have a function $f:\mathbb{R}\to \mathbb{R}$ that is defined by $f(x) = 1$ if $x\in \mathbb{Q}$, and $-\sqrt{3}$ if $x\not\in \mathbb{Q}$.
Using the Integrability Reformulation definition, how would I prove $f(x)=-1$ is not integrable on $[-5,-1]$?
My understanding: Is that I must choose some epsilon and make a partition within $[-5,-1]$ and calculate the upper and lower sums, then subtract the upper and lower sums, but I can't seem to make it work and choose an appropriate epsilon. My textbook is pretty scarce on this topic with no good examples. Thank You in advance.
It doesn't matter what partition you define, every interval in the partition will always have a rational number and an irrational number. So the supremum in each interval in the partition is always $1$ while the infimum is always $-\sqrt{3}$. What can you conclude about the upper and lower sums?