Consider the nonlinear differential operator $a(D^{2}u)$, where x varies in a $C^{2}$-smooth and bounded domain $\Omega \subset \mathbb{R}^{n},u:\Omega \rightarrow \mathbb{R}$ and $D^{2}u$ stands for the Hessian matrix of u. The function $a(x,\xi)$ is supposed to satisfy the Caratheodory condition
Condition ($A^{2}$):There exist three positive constants $\alpha,\gamma,\delta$ such that $\gamma+\delta<1$ and $\bigl|Tr\tau - \alpha (a(x,\xi+\tau)-a(x,\xi))\bigl|^{2} \leq \gamma \|\tau\|_{n^{2}}^{2}+\delta|Tr\tau|^{2}$ $a(x,0)=0$
I need to prove the following
Let a(x,$\xi$) satisfy the condition $(A^{2})$. Then the function $\xi \rightarrow a(x,\xi)$ is differentiable almost everywhere in $\mathbb{R}^{n^{2}}$. Denoting $a^{ij}(x,\xi)=\partial a/ \partial \xi_{ij}(x,\xi), \; i,j=1,...,n$ then $a^{ij} \in L^{\infty}(\Omega\times\mathbb{R}^{n^{2}})$
The function in question is Lipschitz continuous and it follows from Rademacher theorem that it is differentiable a.e. in $\mathbb{R}^{n^{2}}$. My question is why its partial derivatives are bounded and linear.