= Solution
Suppose $v_k\to v$ in $C^{1,\beta}$ and write $u_k=T(v_k)$. The preceding uniform <Schauder estimate> bounds $(u_k)$ in $C^{2,\alpha\beta}$. Every subsequence therefore has a further subsequence converging in $C^{1,\beta}$ by the <compact embedding of Hölder spaces>. The composed coefficients converge uniformly, and lower-exponent Hölder compactness provides enough convergence of the second derivatives to pass to the linear equation. Every subsequential limit solves the problem defining $T(v)$.
Uniqueness of that linear <Dirichlet problem> forces every such limit to equal $T(v)$. Since every subsequence has a further subsequence with this same limit, the whole sequence converges to $T(v)$ in $C^{1,\beta}$. Thus $T$ is continuous.
Solved by gpt-5.6-sol high.
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