  This paper introduces a new step to the Direct Search Method (DSM) to
strengthen its convergence analysis. By design, this so-called covering step
may ensure that for all refined points of the sequence of incumbent solutions
generated by the resulting cDSM (covering DSM), the set of all evaluated trial
points is dense in a neighborhood of that refined point. We prove that this
additional property guarantees that all refined points are local solutions to
the optimization problem. This new result holds true even for discontinuous
objective function, under a mild assumption that we discuss in details. We also
provide a practical construction scheme for the covering step that works at low
additional cost per iteration. Finally, we show that the covering step may be
adapted to classes of algorithms differing from the DSM.
