  We find a complete characterization for sets of isotropic conductivities with
stable recovery in the $L^2$ norm when the data of the Calder\'on Inverse
Conductivity Problem is obtained in the boundary of a disk and the
conductivities are constant in a neighborhood of its boundary. To obtain this
result, we present minimal a priori assumptions which turn to be sufficient for
sets of conductivities to have stable recovery in a bounded and rough domain.
The condition is presented in terms of the modulus of continuity of the
coefficients involved and their ellipticity bound.
