In this article, we propose a novel family of higher order sliding mode (HOSM) controllers with discontinuous integral term for a class of single-input-single-output (SISO) nonlinear systems. They belong to a recently introduced class of controllers, named continuous HOSM since they are able to compensate exactly and in finite-time perturbations/uncertainties using a continuous control signal. They are obtained by introducing a discontinuous integral action at the input of the plant. Here, continuously differentiable Lyapunov functions are used to rigorously prove the stability of the closed-loop system. We also investigate an approximation of this strategy, where a continuous integral term replaces the discontinuous one. The robustness properties of the closed-loop controller with continuous and discontinuous integral term are studied.