Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 1 / 12D / i / Solution

For $x>0$, the function is a composition and product of continuous functions. At zero,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 this function is Riemann integrable. The vanishing factor $x$ controls the infinitely rapid oscillation at the endpoint.