= Solution
Let $M$ bound the piecewise <Hölder norms>, and let $B=\|f\|_\infty$. At scale $j$, classify a <wavelet> as bad when its localized <support of a function> meets a discontinuity, and as good otherwise. <Compact support> and bounded overlap imply at most $C_0K$ bad <wavelets> per scale and at most $C_12^j$ total <wavelets>. A good support lies in a single <smooth> piece, so the preceding <wavelet coefficient decay for Hölder functions> gives $|c_{j,n}|\le C M2^{-j(\alpha+1/2)}$. The permitted bounded-function estimate gives $|c_{j,n}|\le C B2^{-j/2}$ for bad coefficients.
Consider a specific <N-term approximation>: retain all coarse <scaling functions> and all <wavelets> through scale $J$, and also retain the bad coefficients at scales $J<j\le L$, where $L=\lceil2\alpha J\rceil$. Since $\alpha>1/2$, $L\ge J$ for large $J$. Its term count is bounded by
$$
N_J\le C_2 2^J+C_3K(L-J)+C_4=O(2^J+KJ).
$$
By the <Parseval identity>, its squared <L2 norm> error is the sum of omitted squared coefficients. The good tail satisfies
$$
\sum_{j>J}\sum_{n\ \mathrm{good}}|c_{j,n}|^2\le C M^2\sum_{j>J}2^j2^{-2j(\alpha+1/2)}\le C'M^22^{-2\alpha J},
$$
and the remaining bad tail satisfies
$$
\sum_{j>L}\sum_{n\ \mathrm{bad}}|c_{j,n}|^2\le C K B^2\sum_{j>L}2^{-j}\le C'KB^22^{-L}\le C'KB^22^{-2\alpha J}.
$$
Choose $J=\lfloor\log_2(N/(2C_2'))\rfloor$ with a fixed sufficiently large $C_2'$; then the $O(KJ)$ extra terms fit the remaining budget for all sufficiently large $N$, while $2^J$ is comparable to $N$. Padding to $N$ terms does not increase the error. The <best N-term approximation> is at least as accurate as this constructed selection, so
$$
\boxed{\epsilon_n(N,f)=\|f-f_N^{\mathrm{nonlin}}\|_2^2=O(N^{-2\alpha}).}
$$
The constant may depend on the finite number of jumps, the piecewise <Hölder norms>, $B$ and the fixed <wavelet> family. Values exactly at the jumps are immaterial in $L^2$. The improvement comes from retaining only a bounded number of jump-crossing coefficients at each finer scale, as in <best N-term wavelet approximation of piecewise Hölder functions>.
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