We consider the splitting of a reconnecting current sheet into MHD discontinuities, which is observed in many numerical simulations of the magnetic reconnection process. We suppose that the splitting takes place as a consequence of non-evolutionarity of the reconnecting current sheet as a onedimensional discontinuity. This means that the problem of the time evolution of its small perturbations does not have a unique solution. Since a physical problem must always have a unique solution, a non-evolutionary discontinuity cannot exist in a real plasma, and splits into evolutionary discontinuities. Solving the linear MHD equations inside and outside the sheet, we show that for large enough plasma conductivity, certain small perturbations interact with the sheet as with a discontinuity. On the basis of the non-evolutionarity criterion, with respect to these perturbations, we obtain a condition on the flow velocity at the sheet surface, under which the splitting takes place.