Pointwise semi-Lipschitz functions and Banach–Stone theorems

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 function spaces, which is quite flexible and can be applied to many spaces of continuous functions over metric and asymmetric spaces.