Solution
= Solution
For $x>0$, the function is a composition and product of <continuous functions>. At zero,
$$
|x\sin(1/x)-0|\leq x\longrightarrow0.
$$
Thus it is continuous on the whole closed interval, including its assigned value at zero. The continuous-function proof in the introductory solution applies, so \b[this function is Riemann integrable]. The vanishing factor $x$ controls the infinitely rapid oscillation at the endpoint.