Abstract
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.
Highlights
Exception sets for the regularity of a function are encountered when considering functions f : I → R defined on an interval I that have a certain property when restricted to a subset E ⊆ I but not on the whole of I
The property which will be of our interest is the one of Lipschitz continuity and our work is motivated by the desire to generalize the notion of ‘piecewise Lipschitz continuity’ to the multidimensional case
Classical generalizations of Definition 1 to higher dimensions require the desired property on elements of a polytopal, polyhedral or simplicial subdivision of the domain. Variants of this procedure are well known for defining the class of piecewise linear or piecewise differentiable functions, see e.g., [26, 1.4, Ch.1], [28, Section 2.2] or [32, Section 3.9]
Summary
Exception sets for the regularity of a function are encountered when considering functions f : I → R defined on an interval I that have a certain property (e.g. continuity, differentiability) when restricted to a subset E ⊆ I but not on the whole of I. Classical generalizations of Definition 1 to higher dimensions require the desired property on elements of a polytopal, polyhedral or simplicial subdivision of the domain Variants of this procedure are well known for defining the class of piecewise linear (pl) or piecewise differentiable (pdiff) functions, see e.g., [26, 1.4, Ch.1], [28, Section 2.2] or [32, Section 3.9]. The task to determine suitable exception sets for Lipschitz functions with respect to the intrinsic metric should not be mixed with the— related—problem of finding sets R such that functions defined on the complement Rc and belonging to a certain regularity class there may be extended to the whole space. It opens pathways to generalizing results in many applied fields, where concepts of piecewise Lipschitz continuous functions have already been used, such as image processing [5], uncertain input data problems [13], optimal control [14], stochastic differential equations [18], information processing [6, 25], machine learning [4,30], dynamical systems [31], shape-from-shading problems [33]
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