  We study the fundamental properties of pointwise semi-Lipschitz functions
between asymmetric spaces, which are the natural asymmetric counterpart of
pointwise Lipschitz functions. We also study the influence that partial
symmetries of a given space may have on the behavior of pointwise
semi-Lipschitz functions defined on it. Furthermore, we are interested in
characterizing the pointwise semi-Lipschitz structure of an asymmetric space in
terms of real-valued pointwise semi-Lipschitz functions defined on it. By using
two algebras of functions naturally associated to our spaces of pointwise
real-valued semi-Lipschitz functions, we are able to provide two Banach-Stone
type results in this context. In fact, these results are obtained as
consequences of a general Banach-Stone type theorem of topological nature,
stated for abstract functional spaces, which is quite flexible and can be
applied to many spaces of continuous functions over metric and asymmetric
spaces.
