We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body n when the so-called Dirichlet-to-Neumann map is locally given on a non empty portion of the boundary . We extend results of uniqueness and stability at the boundary, obtained by the same authors in SIAM J. Math. Anal. 33:153-171, where the Dirichlet-to-Neumann map was given on all of instead. We also obtain a pointwise stability result at the boundary among the class of conductivities which are continuous at some point y. Our arguments also apply when the local Neumann-to-Dirichlet map is available.