= Solution
Let $f$ be continuous on $[0,1]$. Compactness makes it uniformly continuous. Given $\varepsilon>0$, choose $\delta>0$ such that
$$
|x-y|<\delta\quad\Longrightarrow\quad
|f(x)-f(y)|<\varepsilon.
$$
Choose a partition with mesh smaller than $\delta$. The oscillation $M_j-m_j$ on every subinterval is then at most $\varepsilon$, so
$$
U(f,P)-L(f,P)
=\sum_j(M_j-m_j)(x_j-x_{j-1})
\leq\varepsilon.
$$
The <Riemann integrability criterion> proves that every <continuous function> on $[0,1]$ is Riemann integrable.
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