  We consider criteria for the differentiability of functions with continuous
Laplacian on the Sierpinski Gasket and its higher-dimensional variants $SG_N$,
$N>3$, proving results that generalize those of Teplyaev. When $SG_N$ is
equipped with the standard Dirichlet form and measure $\mu$ we show there is a
full $\mu$-measure set on which continuity of the Laplacian implies existence
of the gradient $\nabla u$, and that this set is not all of $SG_N$. We also
show there is a class of non-uniform measures on the usual Sierpinski Gasket
with the property that continuity of the Laplacian implies the gradient exists
and is continuous everywhere, in sharp contrast to the case with the standard
measure.
