The formulation of inverse problems in the continuum eliminates discretization errors and allows for the exact incorporation of priors. In this paper, we formulate a continuous-domain inverse problem over a search space of continuous and piecewise-linear functions parameterized by box splines. We present a numerical framework to solve those inverse problems with total variation (TV) or its Hessian-based extension (HTV) as regularizers. We show that the box-spline basis allows for exact and efficient convolution-based expressions for both TV and HTV. Our optimization strategy relies on a multiresolution scheme whereby we progressively refine the solution until its cost stabilizes. We test our framework on linear inverse problems and demonstrate its ability to effectively reach a stage beyond which the refinement of the search space no longer decreases the optimization cost.