Everywhere I've tried to look, the only two common examples of non-Riemann integrable functions are unbounded functions or Dirichlet function.
What are some examples of non-Riemann integrable functions besides those functions?
I assume for such non-Riemann integrable functions, the lower integral and the upper integral must exist but are unequal. (Because the lower integral and upper integral both either exist or don't, together and since I have removed unbounded functions from the picture, both integrals exist but are unequal.)
I'd really appreciate multiple examples, if that's possible, of functions that have unequal lower and upper integral.
Best Answer
You can take, for instance$$\begin{array}{rccc}f\colon&[0,1]&\longrightarrow&\Bbb R\\&x&\mapsto&\begin{cases}x&\text{ if }x\in\Bbb Q\\0&\text{ if }x\notin\Bbb Q.\end{cases}\end{array}$$It is bounded. And it is not Riemann integrable since it is discontinuous at every point of $(0,1]$, and this is is uncountable. Another possibility would be$$\begin{array}{rccc}g\colon&[0,1]&\longrightarrow&\Bbb R\\&x&\mapsto&\begin{cases}x&\text{ if }x\in\Bbb Q\\-x&\text{ if }x\notin\Bbb Q.\end{cases}\end{array}$$