Continuous sensitivity equation methods have been applied to a variety of applications ranging from optimal design, to fast algorithms in computational fluid dynamics to the quantification of uncertainty. In order to make use of these methods for interface problems, one needs fast and accurate numerical methods for computing sensitivities for problems defined by partial differential equations with solutions that have spatial discontinuities such as shocks and interfaces. In this paper we develop a discontinuous Petrov-Galerkin finite element scheme for solving the sensitivity equation resulting from a 1D interface problem. The 1D example is sufficient to motivate the theoretical and computational issues that arise when one derives the corresponding boundary-value problem for the sensitivities. In particular, the sensitivity boundary-value problem must be formulated in a very weak sense, and the resulting variational problem provides a natural framework for developing and analyzing numerical schemes. Numerical examples are presented to illustrate the benefits of this approach.