  In the present work, for $\mathbb{X}$ a Banach space, the notion of piecewise
continuous $\mathbb{Z}$-almost automorphic functions with values in finite
dimensional spaces is extended to piecewise continuous $\mathbb{Z}$-almost
automorphic functions with values in $\mathbb{X}$. Several properties of this
class of functions are provided, in particular it is shown that if $\mathbb{X}$
is a Banach algebra, then this class of functions constitute also a Banach
algebra; furthermore, using the theory of $\mathbb{Z}$-almost automorphic
functions, a new characterization of compact almost automorphic functions is
given. As consequences, with the help of $\mathbb{Z}$-almost automorphic
functions, it is presented a simple proof of the characterization of almost
automorphic sequences by compact almost automorphic functions; the method
permits us to give explicit examples of compact almost automorphic functions
which are not almost periodic. Also, using the theory developed here, it is
shown that almost automorphic solutions of differential equations with
piecewise constant argument are in fact compact almost automorphic. Finally, it
is proved that the classical solution of the $1D$ heat equation with continuous
$\mathbb{Z}$-almost automorphic source is also continuous $\mathbb{Z}$-almost
automorphic; furthermore, we comment applications to the existence and
uniqueness of the asymptotically continuous $\mathbb{Z}$-almost automorphic
mild solution to abstract integro-differential equations with nonlocal initial
conditions.
