= Spike obstruction to density-power differentiability
A perturbation can have negligible <Hellinger distance> but a large <integral> of its density's fourth power. At the uniform density on $[0,1]$, let $b(s)=6s(1-s)$ on $[0,1]$ and zero elsewhere, and let $f_t(u)=1-t^4+t^{-2}b(u/t^6)$. Then $\int f_t=1$ and $\int(\sqrt{f_t}-1)^2\leq2t^4=o(t^2)$, so the <score function> is zero. However $\int f_t^4\geq72/(35t^2)$. Thus the <density fourth-power functional> is not continuous along this <differentiable-in-quadratic-mean path>, despite a bounded baseline and individually bounded nearby densities.
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