Population quantiles minimize check loss (${\mathrm{argmin}}_q\mathbb{E}{\rho }_{\tau }(Y-q)=F^{-1}(\tau )$)

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.