  The junction condition across a singular surface in general relativity,
formulated by Israel, has double covariance. In this paper, a general
perturbation scheme of the junction condition around an arbitrary background is
given in a doubly covariant way. After that, as an application of the general
scheme, we consider perturbation of the junction condition around a background
with the symmetry of a $(D-2)$-dimensional constant curvature space, where $D$
is the dimensionality of the spacetime. The perturbed junction condition is
written in terms of doubly gauge-invariant variables only. Since the symmetric
background includes cosmological solutions in the brane-world as a special
case, the doubly gauge-invariant junction condition can be used as basic
equations for perturbations in the brane-world cosmology.
