Abstract This paper derives convergence results for a class of nonnegative absolutely continuous functions that satisfy some differential inequality. These results are shown to be advantageous in studying the behavior at infinity of a class of unboundedly perturbed continuous and discontinuous time-varying systems that is frequently encountered in Lyapunov theory. In particular, sufficient conditions are given to ensure the positivity, the asymptotic stability and the instability of solutions of unidimensional and multidimensional systems. To illustrate the proposed results, several examples are simulated.