= Population quantiles minimize check loss
{title2=$\operatorname*{argmin}_q\mathbb E\rho_\tau(Y-q)=F^{-1}(\tau)$}
For an integrable response with continuous strictly increasing <distribution function> $F$, the <population risk> for the <check loss> has derivative $F(q)-\tau$. Bounded difference quotients justify differentiation by <dominated convergence>. Integrating the derivative between the unique <quantile> and another location proves strict optimality. This proves <identifiability> without requiring a positive density.
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