 Solution

No. Enumerate the rationals in $[0,1]$ asand defineThe function is nonnegative and unbounded, while it is identically zero on $[0,1]\setminus A$. For any $r>0$, however, $f_r$ is positive on the dense set of rationals and zero on the dense set of irrationals. Every interval therefore has positive oscillation, and the upper and lower Darboux integrals differ. Thus even the truncations need not be Riemann integrable.