need example a Riemann-integrable function is not continuous 
every continuous function is Riemann integrable,continuity is certainly not necessary.
I dont know anything about measure.

Define $f$ to be identically $0$ on $\Bbb{R}$, except that $f(0) = 1$. To see that this is Riemann integrable, note that the lower sums are all $0$ (suppose we're integrating on $[-1, 1]$, for clarity). But the upper sums can be made arbitrary small, by choosing small intervals around $0$.

More generally, $f$ is Riemann integrable if and only if the set of points of discontinuities has Lebesgue measure $0$; so something like Thomae's function, which is discontinuous on $\Bbb{Q}$ is still Riemann-integrable.
