= Solution
For a partition
$$
P:\quad 0=x_0<x_1<\cdots<x_m=1,
$$
put
$$
M_j=\sup_{[x_{j-1},x_j]}f,
\qquad
m_j=\inf_{[x_{j-1},x_j]}f.
$$
The upper and lower Darboux sums are
$$
U(f,P)=\sum_{j=1}^mM_j(x_j-x_{j-1}),
\qquad
L(f,P)=\sum_{j=1}^mm_j(x_j-x_{j-1}).
$$
The <upper and lower Darboux integrals> are
$$
\overline{\int_0^1}f=\inf_PU(f,P),
\qquad
\underline{\int_0^1}f=\sup_PL(f,P).
$$
The <bounded function> is Riemann integrable when these two numbers agree; their common value is its <integral>.
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