I'm seeking a function that is non-Riemann integrable yet Lebesgue integrable. Everyone seems to illustrate this phenomenon with the indicator function applied to rational numbers.
Is there another such function... perhaps one that would be of greater interest and background to students... one that can be (approximately) plotted?
We need an uncountable number of singularities. A function such as $f(x) = \frac{1}{\sin 1/x}$ on $[0,1]$ (plotted) has an infinite number of discontinuities, but these can be counted.
