Almost all publications dealing with the stability analysis of nonlinear systems seem to require tough continuity conditions from the differential equations. They may end with nice results, yet for the real-world systems this may also imply that any discontinuity may put in danger the safety of operation. Using and further developing some early results of LaSalle and carefully reviewing the delicate procedure of limit, and then also making use of such apparently abstract concepts as Cantor set, our research managed to eliminate those continuity conditions and thus to improve the guarantee of safe operation of real-world systems. Moreover, because in most cases, the derivative of the Lyapunov function is at most positive semidefinite, the usual analysis can only determine the ultimate behavior of a reduced number of state-variables. Instead, the new analysis allows knowing the ultimate behavior of all state variables.