A continous function on [a,b] is always Riemann integrable. Since is continous it is bounded by extreme value theorem. Furthermore, continous functions on compact (in particular closed) sets are uniformly continous. To prove Riemann integrability we need to show where is a partition of . Given any there is a such that . Pick a partition such that . Then assumes its maximum and minimum on each interval and so , and this quantity can be made arbitrary small.