Stability of asymptotically stable sets for continuous functions

Let K be the class of nonempty compact subsets of I = [0, 1], and K∗ consist of the nonempty closed subsets of K. We study the map A:C(I, I )→K∗ defined so that A(f ) is the closure of the collection of asymptotically stable sets of f . While we find that A is not continuous, we do get much more positive results when studying the semicontinuity of A. In particular, A is lower semicontinuous at a residual set of functions in C(I, I ), so that A is continuous if we restrict our map to a residual subset of C(I, I ).  2002 Elsevier Science (USA). All rights reserved