  We establish sufficient conditions for perfect simulation of chains of
infinite order on a countable alphabet. The new assumption, localized
continuity, is formalized with the help of the notion of context trees, and
includes the traditional continuous case, probabilistic context trees and
discontinuous kernels. Since our assumptions are more refined than uniform
continuity, our algorithms perfectly simulate continuous chains faster than the
existing algorithms of the literature. We provide several illustrative
examples.
