= Solution
Fix nested <relatively compact subsets> of $\Omega$ and let $M$ bound the <gradient> on the larger one. The <fundamental theorem of calculus along a line segment> gives $|w_h|\leq M$ on the smaller one. Part (i) gives coefficient bounds independent of small $h$, so the <De Giorgi-Nash-Moser theorem> gives uniform interior <Hölder continuity> estimates for these <weak solutions>. Since $Du$ is <continuous>, the <difference quotients> $w_h$ converge locally uniformly to $D_\ell u$. Passing to the limit in the <Hölder seminorm> estimate, and repeating for every coordinate, yields
$$
\boxed{u\in C^{1,\alpha}\text{ on each compactly contained subdomain, for some }\alpha>0.}
$$
\b[The first derivatives are locally Hölder continuous.] Here the initially obtained exponent depends on $n$ and the local <gradient> bound. This is the interior interpretation of the printed regularity conclusion; a uniform global estimate does not follow from $u\in C^1(\Omega)$ alone.
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