Boundary differentiability for inhomogeneous infinity Laplace equations

We study the boundary regularity of the solutions to inhomogeneous infinity Laplace equations. We prove that if $u\in C(\bar{\Omega})$ is a viscosity solution to $\Delta_{\infty}u:=\sum_{i,j=1}^n u_{x_i}u_{x_j}u_{x_ix_j}=f$ with $f\in C(\Omega)\cap L^{\infty}(\Omega)$ and for $x_0\in \partial\Omega$ both $\partial\Omega$ and $g:=u|_{\partial\Omega}$ are differentiable at $x_0$, then u is differentiable at $x_0$