Discontinuities in regression functions can reveal important insights. In many contexts, like geographic settings, such discontinuities are multivariate and unknown a priori. We propose a non-parametric regression method that estimates the location and size of discontinuities by segmenting the regression surface. This estimator is based on a convex relaxation of the Mumford-Shah functional, for which we establish identification and convergence. We use it to show that an internet shutdown in India resulted in a reduction of economic activity by 25Ð35%, greatly surpassing previous estimates and shedding new light on the true cost of such shutdowns for digital economies globally.