This article analyzes the performance of the Continuous Basis Pursuit (C-BP) method for sparse super-resolution. The C-BP has been recently proposed by Ekanadham, Tranchina and Simoncelli as a refined discretization scheme for the recovery of spikes in inverse problems regularization. One of the most well known discretization scheme, the Basis Pursuit (BP, also known as Lasso) makes use of a finite dimensional l1 norm on a grid. In contrast, the C-BP uses a linear interpolation of the spikes positions to enable the recovery of spikes between grid points. When the sought-after solution is constrained to be positive, a remarkable feature of this approach is that it retains the convexity of the initial l1 problem. The present paper shows how the C-BP is able to recover the spikes locations with sub-grid accuracy in the favorable case. We also prove that this regime generally breaks when the grid is too thin, and we describe precisely the artifacts that appear: each spike is approximated by a pair of Dirac masses. We show numerical illustrations of these phenomena, and evaluate numerically the validity of the technical assumptions of our analysis.