= Solution
For a partition $P$, write $U(f,P)$ and $L(f,P)$ for the <upper Darboux sum> and <lower Darboux sum>. The <Riemann integrability criterion> for bounded $f$ is that for every $\varepsilon>0$ some partition satisfies $U(f,P)-L(f,P)<\varepsilon$. Thus if the gaps for $D_n$ tend to zero, the <Riemann integrability criterion> immediately proves that $f$ is <Riemann integrable>.
Conversely, suppose $f$ is <Riemann integrable>, and choose a fixed partition $P$ with $U(f,P)-L(f,P)<\varepsilon/2$. Let $r$ be the number of its interior division points and choose $M\geq1$ with $|f|\leq M$. Call a cell of $D_n$ bad when its interior contains a division point of $P$. There are at most $r$ bad cells, and their total length is at most $r/n$.
Every other cell lies in a single cell of $P$, so its oscillation is no larger than that of the containing cell. Summing the contributions of these good cells gives at most $U(f,P)-L(f,P)$, while each bad cell has oscillation at most $2M$. Therefore the <finite bad-cell estimate for Darboux sums> gives
$$
0\leq U(f,D_n)-L(f,D_n)\leq U(f,P)-L(f,P)+\frac{2Mr}{n}.
$$
For large enough $n$, the right-hand side is less than $\varepsilon$. This proves the <uniform-mesh Darboux criterion>
$$
\boxed{f\text{ is Riemann integrable}\quad\Longleftrightarrow\quad U(f,D_n)-L(f,D_n)\longrightarrow0.}
$$
This argument does not assume that the partitions $D_n$ are nested; in general they are not.
For the composition, $f$ takes values in $[-M,M]$. Since $g$ is continuously differentiable, the <extreme value theorem> bounds both $|g|$ and $|g'|$ on that interval. In particular, let $K=\sup_{[-M,M]}|g'|<\infty$. The <mean value theorem> states that a function continuous on the interval between $s,t$ and differentiable in its interior satisfies $g(s)-g(t)=g'(\xi)(s-t)$ for some intermediate $\xi$. Hence
$$
|g(s)-g(t)|\leq K|s-t|\qquad(s,t\in[-M,M]).
$$
On each partition cell, this bounds the oscillation of $g\circ f$ by $K$ times the oscillation of $f$, whether or not the local extrema are attained. It follows that
$$
U(g\circ f,D_n)-L(g\circ f,D_n)\leq K\bigl(U(f,D_n)-L(f,D_n)\bigr)\longrightarrow0.
$$
The composition is bounded, so the proved <uniform-mesh Darboux criterion> applies. Thus \b[$g\circ f$ is Riemann integrable]. The reusable fact is <Lipschitz composition preserves Riemann integrability>; continuous differentiability supplies the required bound on the range of $f$.
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