We provide a Lyapunov-based design of High-Order Sliding Mode (HOSM) Control of arbitrary order for a class of Single-Input-Single-Output uncertain nonlinear systems. First we provide a general method to design degree 0 homogeneous controllers based on a homogeneous Control Lyapunov Function (CLF). Then, for a perturbed chain of integrators we construct a smooth and homogeneous CLF and use it to design a family of homogeneous and discontinuous control laws. The controller drives in finite time the trajectories of the system to a sliding surface of arbitrary relative degree, and keeps the trajectories on this manifold despite persistently acting matched perturbations, so that a sliding mode is established. We obtain a large and simply parametrized family of "usual" discontinuous and quasi-continuous HOSM controllers of nested and polynomial type, that share the robustness and accuracy properties of the existing ones. (C) 2017 Elsevier Ltd. All rights reserved.