= Solution
The <continuous approximation of a Riemann-integrable function> follows here by choosing partitions $P_n$ with upper-minus-lower sum below $1/n$. On each subinterval choose the midpoint of the infimum and supremum to form a step <function> $s_n$. Then
$$
\int_a^b|g-s_n|<\frac1n.
$$
Replace each of the finitely many jumps of $s_n$ by a linear transition on intervals of sufficiently small total length, obtaining a continuous $\phi_n$ with $\int|s_n-\phi_n|<1/n$. Hence $\int|g-\phi_n|<2/n$, and uniformly for every subinterval $[\alpha,\beta]$,
$$
\boxed{\left|\int_\alpha^\beta(g-\phi_n)\right|\leq\int_a^b|g-\phi_n|\to0.}
$$
Back to article page
