 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 $s_n$. ThenReplace each of the finitely many jumps of $s_n$ by a linear transition on intervals of sufficiently small total length, obtaining a continuous ${\varphi }_n$ with $\int |s_n-{\varphi }_n|<1/n$. Hence $\int |g-{\varphi }_n|<2/n$, and uniformly for every subinterval $[\alpha ,\beta ]$,