A diagonal ridge fails the mixed fourth-derivative hypothesis

Fix an interior $y\in (0,1)$ and write $g(h)=f(y+h,y)=\mathrm{\mid }h{\mathrm{\mid }}^3$ for sufficiently small positive and negative $h$. By [F4]–[F5], on the two sides $g^{\mathrm{\prime }}(h)=3h\mathrm{\mid }h\mathrm{\mid }$ and $g^{\mathrm{\prime }\mathrm{\prime }}(h)=6\mathrm{\mid }h\mathrm{\mid }$ for $h\mathrm{\ne }0$. The difference quotients at zero give $g^{\mathrm{\prime }}(0)=0$ and $g^{\mathrm{\prime }\mathrm{\prime }}(0)={\lim }_{h\to 0}3\mathrm{\mid }h\mathrm{\mid }=0$. The derivative of $g^{\mathrm{\prime }\mathrm{\prime }}$ at zero would require ${\lim }_{h\to 0}(g^{\mathrm{\prime }\mathrm{\prime }}(h)-g^{\mathrm{\prime }\mathrm{\prime }}(0))\mathrm{/}h={\lim }_{h\to 0}6\mathrm{\mid }h\mathrm{\mid }\mathrm{/}h$, whose right-hand value is $6$ and left-hand value is $-6$. Thus $g^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(0)$ does not exist, so $g^{(4)}(0)$ cannot exist. By [F3], the required derivative therefore fails at the interior diagonal point $(y,y)$.

Since [F1] requires $D_{\{1\}}f={\mathrm{\partial }}_x^{4}f$ to exist continuously everywhere, the failure in step 1.1 violates a stated hypothesis of the rate theorem [F2]. The theorem makes no such rate claim for this ridge.

For every fixed $y$, the same formulas give ${\mathrm{\partial }}_x^{2}f(x,y)=6\mathrm{\mid }x-y\mathrm{\mid }$, including its value zero on the diagonal; the same one-sided calculations apply at $x=0,1$. By symmetry ${\mathrm{\partial }}_y^{2}f(x,y)=6\mathrm{\mid }x-y\mathrm{\mid }$ as well. These partials are continuous and bounded by $6$ on the square since $\mathrm{\mid }x-y\mathrm{\mid }\le 1$, illustrating that lower-order smoothness does not imply the fourth-order condition used here.