C-boundary regularity of planar infinity harmonic functions

We prove that if Ω ⊂ R is a bounded domain with C-boundary and g ∈ C(R), then any viscosity solution u ∈ C(Ω) of the infinity Laplacian equation (1.1) is C(Ω). The interior C and C-regularity of u in dimension two has been proved by Savin [20] and Evans-Savin [15] respectively. We also show that for any n ≥ 3, if Ω ⊂ R is a bounded domain with C-boundary and g ∈ C(R), then the solution u of equation (1.1) is differentiable on ∂Ω. This can be viewed as a supplementary result to the much deeper interior differentiability theorem by Evans-Smart [16, 17].