Upper Semicontinuity of Morse Sets of a Discretization of a Delay-diierential Equation:a Complete Solution

In this paper, we consider a discrete delay problem with negative feedback _ x(t) = f(x(t); x(t?1)) along with a certain family of time discretizations with stepsize 1=n. In the original problem, the attractor admits a Morse decomposition. We proved in G,H] that the discretized problems have global attractors. It was proved in G,M] that such attractors also admit Morse decompositions. In G,H] we proved certain continuity results about the individual Morse sets, including that if f(x; y) = f(y), then the individual Morse sets are upper semicontinuous at n = 1. In this paper we extend this result to the general case; that is we prove for general f(x; y) with negative feedback that the Morse sets are upper semicontinuous.