  In this paper we derive a propagation of smallness result for a scalar second
elliptic equation in divergence form whose leading order coefficients are
Lipschitz continuous on two sides of a $C^2$ hypersurface that crosses the
domain, but may have jumps across this hypersurface. Our propagation of
smallness result is in the most general form regarding the locations of
domains, which may intersect the interface of discontinuity. At the end, we
also list some consequences of the propagation of smallness result, including
stability results for the associated Cauchy problem, a propagation of smallness
result from sets of positive measure, and a quantitative Runge approximation
property.
