Brief comments for the continuation method by A. F. Filippov for solution continuation on a discontinuity set

The conventional existence-uniqueness theorems are not applicable for differential equations with right hand sides as discontinuous state functions. This is the case for the systems with discontinuous controls and sliding modes, when state trajectories belong to discontinuity surfaces. Many authors offered their methods of deriving sliding mode equations, or solution continuations on the discontinuity surfaces Due to uncertainties of right hand sides of the original equations the proposed methods led to different solutions. These methods are compared, the reasons of ambiguity are discussed in the paper. It is assumed that any solutions is under the umbrella of the method proposed by A. F. Filippov.