Thus the problem comes to approximating our given function by one which satisfies a Lipschitz condition, still preserving the derivative of course. This problem is solved in the second section by using an averaging technique which was developed by Kurzweil [3]. Unfortunately, while the convolution provides a close approximation for the derivative, the Kurzweil averaging only allows for a one-sided approximation of the derivative. However, this is enough for our purposes, and in the third section we apply these results to the converse Lyapunov problem. In the final section of this paper we apply these results to obtain a topological version of the Hirsch-Cairns smoothing theorem. We owe special gratitude to Charles C. Pugh for his careful reading of the manuscript, especially ?2. Because of his diligence, the reader will be spared many ambiguities and typographical errors, and in addition one rather annoying technical error which appeared in the original manuscript. 1. Smoothing Lipschitzian Functions. Let M be a paracompact Cco manifold, and let X be a continuous nonsingular vector field on M whose trajectories are defined for all time and depend uniquely upon their initial conditions. Suppose that f: M -- R is a given continuous function such that the derivative Xf is defined and