= Solution
For a bounded real function on $[0,1]$, take a partition $P$ with nodes $0=x_0<\cdots<x_N=1$. Its <lower Darboux sum> and <upper Darboux sum> are
$$
L(f,P)=\sum_j\inf_{[x_{j-1},x_j]}f\,(x_j-x_{j-1}),\qquad
U(f,P)=\sum_j\sup_{[x_{j-1},x_j]}f\,(x_j-x_{j-1}).
$$
The function is <Riemann integrable> when $\sup_P L(f,P)=\inf_P U(f,P)$; their common value is its <Riemann integral>. Equivalently, for every $\eta>0$ some partition has $U-L<\eta$. Indeed a common refinement of nearly extremizing upper and lower partitions proves necessity, while arbitrarily small gaps force equality of the two integrals. This is the <Riemann integrability criterion>.
A <continuous> function on a compact interval is bounded and <uniformly continuous>. Choose a mesh small enough that its oscillation on each subinterval is less than $\eta$. Then $U-L\leq\eta\sum_j(x_j-x_{j-1})=\eta$; using $\eta/2$ first makes the desired inequality strict. Thus \b[every <continuous> function on $[0,1]$ is <Riemann integrable>].
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