I was wondering if people can give me "nice" examples of non-Riemann integrable functions. I know the one about the rationals and irrationals, so-called indicator function (and called something else by a lot of other people), but I was hoping for something a little more natural. I'm trying to see whether $\displaystyle f^{2} \in \mathscr{R}$ entails that $\displaystyle f \in \mathscr{R}$, and I thought that if I had a few examples of non-integrable functions it might help, but there are precious few to be found by a basic Google search.