  We consider a `contracting boundary' of a proper geodesic metric space
consisting of equivalence classes of geodesic rays that behave like geodesics
in a hyperbolic space. We topologize this set via the Gromov product, in
analogy to the topology of the boundary of a hyperbolic space. We show that
when the space is not hyperbolic, quasi-isometries do not necessarily give
homeomorphisms of this boundary. Continuity can fail even when the spaces are
required to be CAT(0). We show this by constructing an explicit example.
