  We study unique continuation over an interface using a stabilized unfitted
finite element method tailored to the conditional stability of the problem. The
interface is approximated using an isoparametric transformation of the
background mesh and the corresponding geometrical error is included in our
error analysis. To counter possible destabilizing effects caused by
non-conformity of the discretization and cope with the interface conditions, we
introduce adapted regularization terms. This allows to derive error estimates
based on conditional stability. Numerical experiments suggest that the presence
of an interface seems to be of minor importance for the continuation of the
solution beyond the data domain. On the other hand, certain convexity
properties of the geometry are crucial as has already been observed for many
other problems without interfaces.
