The dynamical behavior of a map on the unit interval has been the subject of
much contemporary research. In this paper, we will consider the relation between
the continuity of the map cof and cof for some positive integer k, where f
is a continuous map from the unit interval to itself, and ?f is a function which
takes any element of the unit interval to the set of all subsequential limits of the
orbit of x under f Also, it is shown that for any k, the continuity of ?f implies
the equicontinuity of the iterates { f n}.