We construct a piecewise-polynomial interpolant $u \mapsto \Pi u$ for functions $u:\Omega \setminus \Gamma \to \mathbb{R}$, where $\Omega \subset \mathbb{R}^d$ is a Lipschitz polyhedron and $\Gamma \subset \Omega$ is a possibly non-manifold $(d-1)$-dimensional hypersurface. This interpolant enjoys approximation properties in relevant Sobolev norms, as well as a set of additional algebraic properties, namely, $\Pi^2 = \Pi$, and $\Pi$ preserves homogeneous boundary values and jumps of its argument on $\Gamma$. As an application, we obtain a bounded discrete right-inverse of the "jump" operator across $\Gamma$, and an error estimate for a Galerkin scheme to solve a second-order elliptic PDE in $\Omega$ with a prescribed jump across $\Gamma$.

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