How to check if function is a Riemann integrable on $[0,1]$? I don't know how to find lower and upper Darboux sum.
$$f(x) =\left\{ \begin{array}{ll} 1,\text{when}\ \exists \ m,n \in \mathbb{N}, x = \frac{m}{2^n}\\ 0, \text{otherwise} \end{array} \right.$$
Take any subinterval of $[0,1]$. It contains a number of the form $m/2^n$ (just take any $n$ so that the length of the interval is smaller than $2^{-n}$). It also contains a number not of that form, e.g. any irrational number. That means that the supremum of $f$ on any interval is 1, and the infimum is 0.
