 Solution

Let $f$ be continuous on $[0,1]$. Compactness makes it uniformly continuous. Given $\epsilon >0$, choose $\delta >0$ such thatChoose a partition with mesh smaller than $\delta$. The oscillation $M_j-m_j$ on every subinterval is then at most $\epsilon$, soThe Riemann integrability criterion proves that every continuous function on $[0,1]$ is Riemann integrable.