  We discuss the inverse problem of determining the, possibly anisotropic,
conductivity of a body $\Omega\subset\mathbb{R}^{n}$ when the so--called
Dirichlet-to-Neumann map is locally given on a non empty portion $\Gamma$ of
the boundary $\partial\Omega$. We extend results of uniqueness and stability at
the boundary, obtained by the same authors in SIAM J. Math. Anal. 33 (2001),
no. 1, 153--171, where the Dirichlet-to-Neumann map was given on all of
$\partial\Omega$ instead. We also obtain a pointwise stability result at the
boundary among the class of conductivities which are continuous at some point
$y\in\Gamma$. Our arguments also apply when the local Neumann-to-Dirichlet map
is available.
