  Given a dynamical system $(X,f)$, we let $E(X,f)$ denote its Ellis semigroup
and $E(X,f)^* = E(X,f) \setminus \{f^n : n \in \mathbb{N}\}$. We analyze the
Ellis semigroup of a dynamical system having a compact metric countable space
as a phase space. We show that if $(X,f)$ is a dynamical system such that $X$
is a compact metric countable space and every accumulation point $X'$ is
periodic, then either each function of $E(X,f)^*$ is continuous or each
function of $E(X,f)^*$ is discontinuous. We describe an example of a dynamical
system $(X,f)$ where $X$ is a compact metric countable space, the orbit of each
accumulation point is finite and $E(X,f)^*$ contains continuous and
discontinuous functions.
