Well, I would want to construct a partition that minimizes the effect of the discontinuity at

by embedding

in a very small subinterval.

I am using the book "Understanding Analysis" by Stephen Abbott. The book is current; the problem I posted is in a section called "integration functions with discontinuities".

I have found very little on the subject in my google searchers. In the next section, "measure zero" is brought up and the only difference I can tell is that "measure zero" handles the infinitely countable case.