  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 25--35%, greatly surpassing previous estimates and shedding new
light on the true cost of such shutdowns for digital economies globally.
