  We prove that every continuous map acting on the four-dimensional Minkowski
space and preserving light cones in one direction only is either a Poincar\'e
similarity, that is, a product of a Lorentz transformation and a dilation, or
it is of a very special degenerate form. In the presence of the continuity
assumption the main tool in the proof is a basic result from the homotopy
theory of spheres.
