Density fourth-power functional ($\psi (f)=\int f^4$)

Along bounded density tilts, this functional has derivative $4\int f^{4}g=P_f(4f^{3}g)$. At a bounded baseline density its canonical gradient, relative to these regular paths, is $4(f^3-\int f^4)$. A common local bound on nearby densities suffices to obtain the same derivative along arbitrary differentiable-in-quadratic-mean paths in that bounded neighborhood. A bounded baseline alone is insufficient, because of the 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 $\int (\sqrt{f_t}-1{)}^2\le 2t^4=o(t^2)$, so the score function is zero. However $\int f_t^4\ge 72/(35t^2)$. 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.