  We consider a class of functions defined on metric spaces which generalizes
the concept of piecewise Lipschitz continuous functions on an interval or on
polyhedral structures. The study of such functions requires the investigation
of their exception sets where the Lipschitz property fails. The newly
introduced notion of permeability describes sets which are natural exceptions
for Lipschitz continuity in a well-defined sense. One of the main results
states that continuous functions which are intrinsically Lipschitz continuous
outside a permeable set are Lipschitz continuous on the whole domain with
respect to the intrinsic metric. We provide examples of permeable sets in
$\mathbb{R}^d$, which include Lipschitz submanifolds.
