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Let $f_i$ be any sequence of strictly increasing smooth functions that converge to the Sign function, such as $f_i(x) = \tanh(ix)$, and let $g_i$ be defined by $g_i(x,y) = f_i(x+f_i^{-1}(y))$ for $y$s in the range of $f_i$ (e.g., $-1 < y < 1$) and however you like elsewhere (since the problem posits no smoothness or even any continuity conditions on $g$). In short, the "decomposability constraint" doesn't seem to amount to much.