  In this paper we consider a very singular elliptic equation that involves an
anisotropic diffusion operator, including one-Laplacian, and is perturbed by a
$p$-Laplacian-type diffusion operator with $1<p<\infty$. This equation seems
analytically difficult to handle near a facet, the place where the gradient
vanishes. Our main purpose is to prove that weak solutions are continuously
differentiable even across the facet. Here it is of interest to know whether a
gradient is continuous when it is truncated near a facet. To answer this
affirmatively, we consider an approximation problem, and use standard methods
including De Giorgi's truncation and freezing coefficient methods.
