This article treats the effects of initial jump discontinuities of the inputs on solutions of systems represented by nonlinear second-order ordinary differential equations with terms containing differentials of the input function. A methodology for analysis of discontinuities is presented. In some cases, an initial discontinuity of the calculated impulse response cannot be accounted for in the input function. The value of that discontinuity is thus required for the correct solution of such cases. However, that value cannot be obtained unless the solution profile is known. This paradoxical effect is investigated for different systems and validated by the comparison of experimental and simulated data of nonlinear models of flow-level tanks. A few systems are not affected by discontinuities upon linearization. All others, however, retain their maiden character of being affected or not affected by discontinuities upon linearization. Some interesting cases of calculations of time required for emptying flow-level tanks using impulse responses are also discussed.