Super-resolution localization microscopy relies on sparse activation of photo-switchable probes. Such activation, however, introduces limited temporal resolution. High-density imaging overcomes this limitation by allowing several neighboring probes to be activated simultaneously. In this work, we propose an algorithm that incorporates a continuous-domain sparsity prior into the high-density localization problem. We use a Taylor approximation of the PSF, and rely on a fast proximal gradient optimization procedure. Unlike currently available methods that use discrete-domain sparsity priors, our approach does not restrict the estimated locations to a pre-defined sampling grid. Experimental results of simulated and real data demonstrate significant improvement over these methods in terms of accuracy, molecular identification and computational complexity.