We present in this paper a continuous feedback approach to the problem of global strong stabilization, for genuinely nonlinear systems that may not be stabilized, even locally, by any smooth feedback. We describe conditions under which it is possible to prove, while no smooth controllers exist, the existence of continuous state feedback control laws that achieve global strong stability (GSS) in the sense of Kurzweil, The proof is constructive and carried out by developing a machinery, which combines the theory of homogeneous systems with the idea of adding a power integrator, for the explicit construction of globally stabilizing continuous controllers. We then illustrate, by means of examples, how this machinery can be used to overcome the topological obstruction caused by smooth feedback, and hence resulting in new solutions to a variety of open control problems, including global stabilization of an underactuated unstable two degree of freedom mechanical system.