We study the super-resolution (SR) problem of recovering point sources consisting of a collection of isolated and suitably separated spikes from only the low frequency measurements. If the peak separation is above a factor in (1,2) of the Rayleigh length (physical resolution limit), L 1 minimization is guaranteed to recover such sparse signals. However, below such critical length scale, especially the Rayleigh length, the L 1 certificate no longer exists. We show several local properties (local minimum, directional stationarity, and sparsity) of the limit points of minimizing two L 1 based nonconvex penalties, the difference of L 1 and L 1 norms ( L 1 ) and capped L 1 (C L 1 ), subject to the measurement constraints. In one and two dimensional numerical SR examples, the local optimal solutions from difference of convex function algorithms outperform the global L 1 solutions near or below Rayleigh length scales either in