Exception Sets of Intrinsic and Piecewise Lipschitz Functions

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {R}}}^d$$\end{document}Rd, which include Lipschitz submanifolds.


Introduction
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 . The complement = I \ E is then an exception set for the function's property. A particularly common example here is the case of a finite exception set for a function defined on a real interval. Such a finite set partitions the interval I into several parts or pieces, hence leading to the notions of piecewise continuous, differentiable, etc. functions. 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. Consider the following definition: Definition 1 A function f : I → R is piecewise Lipschitz continuous if there exist finitely many points x 1 , . . . , x n ∈ I with x 0 := inf I < x 1 < . . . < x n+1 := sup I , such that f | (x j−1 ,x j ) is Lipschitz continuous for every j = 1, . . . , n + 1.
With this definition, the following result is easily proven (see, for example, Lemma 2.4 in [18]):

Lemma 2 Let I ⊆ R be an interval and let f : I → R be continuous and piecewise Lipschitz continuous. Then f is Lipschitz continuous.
Definition 1 is a reasonable implementation of the concept, although we do not claim that it is universally accepted across the mathematical community. Generalizing this (or a similar) definition to the multidimensional case is far from being unambiguous as many multidimensional concepts coincide in dimension 1: Intervals are precisely the convex subsets of R, but also precisely the star-shaped, connected, pathconnected, arc-connected sets and the polytopes. Multidimensional generalizations of the exception set {x 1 , . . . , x n } in Definition 1 are affine hyperplanes, (finite unions of) submanifolds, finite sets, among others. 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]. However, these classes comprise of continuous functions. A definition of piecewise linear using a subdivison by hyperplanes not implying continuity can be found in the introduction of [8]. A simple extension of the notion 'piecewise Lipschitz continuous', loosely following the ideas in the references above, could be given by the subsequent definition: finitely many open polyhedra 1 P 1 , . . . , P m with P j ∩ P k = ∅ for j = k and M ⊆ n j=1 P j , such that f | P j is Lipschitz continuous for every j = 1, . . . , n.
The definition in [22] (implicit in Assumption 2.2) is similar to Definition 3,but replaces 'open polyhedra' by 'open sets'. In [4], general sets for the P i are allowed, and the resulting difficulties are overcome by considering a 'dispersed' family of functions which means that not too many of them have their Lipschitz-exceptions around the same spot. We will take a path different from Definition 3 by concentrating on a notion which, instead of focussing on the pieces on which the Lipschitz property holds, we emphasize the exception set where the Lipschitz property fails. Related approaches can be found in [19], where the authors consider functions which are Lipschitz on each of two parts of a domain which is split by a C 1,1 manifold. A more general exception set occurs in [13,Section 14.2], where a piecewise Lipschitz continuous function is one which is is defined on a union of domains with Lipschitz boundaries, which is Lipschitz continuous on these subdomains. An even more complex exception set is allowed in [20], where a function is 'piecewise C 2 ', if the C 2 property fails on a closed set of Lebesgue-measure 0. An interesting theorem outside the R d -setting can be found in [7], where it is shown that a specific notion of piecewise Lipschitz continuity follows from a local Lipschitz condition for semi-algebraic functions Q d p → Q p , where Q p are the p-adic numbers.
A guiding principle for what we attempt here will be that a suitable generalization of Lemma 2 should hold. We generalize and extend the approach of [18], who call a function f : is Lipschitz with respect to the intrinsic metric (see Definition 6) on R d \ , and where is a hypersurface, that is, a (d − 1)-dimensional submanifold of R d . They prove a multidimensional version of Lemma 2 under an additional condition on , namely, the condition that we will call finitely permeable (see Definition 10). The task to determine suitable exception sets for Lipschitz functions with respect to the intrinsic metric should not be mixed with the-of course related-problem of finding sets R such that functions defined on the complement R c and belonging to a certain regularity class there may be extended to the whole space. Such removable sets R have been investigated in complex analysis and geometric function theory for a long time, see e.g., [15,36], and, in connection with Lipschitz continuity, [9]. In the early account [1], removable sets are called function theoretic null-sets and are characterized by an extremal distance condition.
The generalization of piecewise Lipschitz continuous functions to Lipschitz continuous functions with respect to the intrinsic metric up to an exception set includes far more functions than when the induced metric on the complement of the exception is used, see Example 8 in Sect. 2 for instance. Also, Lemma 2 does not simply follow from well-known extension theorems, such as the classical ones of Kirszbraun (see [17], [29, p.21]), McShane-Whitney (see [21,34]) or more recent ones such as the one in [23].
We expand the generalization from [18] in many directions. In Sect. 2 we first recall the notion of the intrinsic metric and introduce the concept of permeable and finitely permeable sets. This is done in the general framework of metric spaces. We define intrinsically Lipschitz continuous functions as functions which are Lipschitz continuous with respect to the intrinsic metric on their domain. Our first main result is then Theorem 15, which is a multidimensional version of Lemma 2, where the additional assumption is that the exception set is permeable. No further assumptions are required, in particular, the exception set need not be a manifold. The proof uses transfinite induction and the Cantor-Bendixson theorem.
The notion of permeability is weaker than that of finite permeability, and much weaker than finiteness, and we show that in the 1-dimensional real case it not only is a sufficient condition on a set so that every continuous function which is Lipschitz continuous with exception set is Lipschitz-it is also necessary.
Section 3 is then dedicated to finding large and practically relevant classes of subsets of R d which are permeable. We show in Theorem 31 that every Lipschitz submanifold which is a closed subset of R d is finitely permeable and thus permeable. We discuss further generalizations, instructive examples and counterexamples.
Our research presents a new concept in analysis with already a number of non-trivial results and generalizations. Moreover, 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].

Intrinsic Lipschitz Functions and Permeable Subsets of Metric Spaces
Throughout this section, let (M, d) be a metric space. To begin, we recall some definitions for metric spaces.
The following important lemma is an immediate consequence of Propositions 3.4 and 3.5 in [2]. It states that one can always replace a path in M by an injective one with length at most that of the original path and its image contained in that of the original path.
Note that ρ E is not a proper metric in that it may assume the value infinity. Of course, one could relate ρ E to a proper metric However, we stick to ρ E , as it is the more natural choice and the extended co-domain does not lead to any difficulties. It is readily checked that, if we allow ∞ as the value of a metric, then (E, ρ E ) is a length space.
See [12,Section 7] and [11] for interesting consequences of quasi-convexity in the context of Lipschitz analysis.

Definition 7 (Intrinsically Lipschitz continuous function)
continuous for some L.

In the above cases we call M \ E an exception set (for intrinsic Lipschitz continuity)
of f .

Example 8
Consider the function f : It is readily checked, however, that f is Lipschitz continuous on E = R 2 \{x ∈ R 2 : Thus f is intrinsically Lipschitz continuous with exception set := {x ∈ R 2 :

arg(x) is only locally intrinsically Lipschitz continuous on E.
A classical method for proving Lipschitz continuity of a differentiable function also works for intrinsic Lipschitz continuity: A proof can be found in [18,Lemma 3.6].
It is almost obvious, that a function f : R 2 → R, which is continuous and intrinsically Lipschitz continuous on does not pose a 'hard' barrier, since every straight line connecting two points in R 2 \ has at most one intersection point with and so one can conclude the Lipschitz continuity by approaching from either side (we invite the reader to make this argument rigorous -such a kind of argument will be used also in the proof of Theorem 15).
To make the elementary property of 'not being a hard barrier' precise, we define at this point the notion of permeability.
When the ambient space (M, d) is understood and there is no danger of confusion, we simply say is (finitely) permeable.

Remark 11
A set ⊆ M is (finitely) permeable iff for any x, y ∈ M and every ε > 0 there exists a path γ from x to y in M with (γ ) < ρ M (x, y) + ε and such that {γ (t) : t ∈ [a, b]} ∩ is at most countable (finite). Clearly, every finitely permeable set is permeable.
The notion of permeability is related to that of metrical removability [15, Definition 1.1]: A set ∈ M is metrically removable if for all x, y ∈ M and all ε > 0 there y) it follows that every metrically removable set is finitely permeable.
Lemma 3.7 in [15] states that if M ⊆ R n , then a subset ⊆ M is metrically removable if and only if ρ M = ρ M\ . Therefore, for subsets M of the R n with ρ M = d (i.e., M is a length space), metrical removability corresponds to Definition 10, where 'countable' or 'finite' is replaced by 'empty'.
⊆ M is permeable, then it has no interior point with respect to the original metric d.
Proof Let x be an interior point of and y ∈ M \ {x}. By our assumption there exists a path γ : [0, 1] → M from x to y with finite length. By Lemma 5 there exists an arc η : such that the latter set is uncountable. Since γ was an arbitrary path in M from x to y, it follows that (x, y) = ∅, and therefore We will show in Sect. 3 that all sufficiently regular sub-manifolds of the R d of dimension smaller than d are finitely permeable. Next we show that (finite) permeability transfers to subsets.
Proof Let x, y ∈ M and ε > 0. There exists γ : The 'finitely permeable' case follows from similar considerations.
If we can show that the map f • γ : [0, b] → Y is L-Lipschitz continuous, then we are done, because then Then A is closed since γ is continuous. We start by showing that if C 0 is a connected component of [0, b] \ A (clearly, C 0 is an interval with non-empty interior), then f • γ is L-Lipschitz on C 0 . Indeed, let r , s ∈ C • 0 with r < s. Since [r , s] ∩ A = ∅, the restricted arc γ | [r ,s] is an arc in E and therefore ρ E (γ (r ), γ (s)) ≤ (γ | [r ,s] ) = (s −r ) . Since f is intrinsically L-Lipschitz continuous on E, By the continuity of f , Eq. (1) also holds for r , s ∈ C 0 . Next let C 1 be a connected component of That is, and by the continuity of f , Equation (3) also holds for r , s ∈ C 1 . We proceed by a transfinite induction argument, where the base case has already been dealt with. The induction hypothesis is that for an ordinal α, all ordinals β < α and every connected component we have that f •γ is L-Lipschitz continuous on C β . In order to perform the induction step, we need to show that this property extends to α, that is, for every connected component If α is not a limit ordinal, we can use the same argument as in the step from the ordinal 0 to the ordinal 1. Now assume, that α is a limit ordinal. Let r , s ∈ C • α , r < s. There exists an ordinal β 0 < α such that [r , s] ∩ H β 0 (A) = ∅ : Otherwise, there is an increasing sequence (β n ) n∈N , β n < α for all n, and a sequence (t n ) n∈N

Remark 17
Note that the inequalities in (2) cannot be generalized in a straightforward way to a related notion of "intrinsically Hölder continuous". [18]. In the latter it is assumed that the exception set is a finitely permeable submanifold of R d . Then f is C-Lipschitz on I . This result is obviously also related to our 1-dimensional permeability criterion, Theorem 23, below.

Remark 18 1. Theorem 15 generalizes Lemma 3.6 in
We have two immediate corollaries of Theorem 15: For example, Corollary 20 can be applied for M = R d with the euclidean metric, which is a length space. We will take a deeper look at this example in Sect. 3.

Corollary 19 Let M be a C-quasi-convex space and let
The next result states that, if a function is intrinsically Lipschitz except on a closed permeable set, then the Lipschitz constant does not change when one enlarges the exception set to another permeable set.
By assumption, f is intrinsically Lipschitz continuous on N .
As ⊇ 0 , f is also intrinsically Lipschitz continuous on M \ = N \ ( \ 0 ), and \ 0 is permeable in N , by Proposition 14. Therefore the assertion follows from Theorem 15.
We now show that in the one-dimensional euclidean case the permeable sets are precisely those for which every function is Lipschitz iff it is continuous and intrinsically Lipschitz with exception set . Note that for subsets of R permeability is equivalent to having countable closure. For the "if part" assume to the contrary that has uncountable closure. Then = A ∪ P, where A is countable and P is perfect by the Cantor-Bendixson theorem, [16,Theorem 6.4]. Hence, P contains a homeomorphic copy of the Cantor set by [16,Theorem 6.5]. Let f be the corresponding Cantor staircase function. Then f is continuous and f is constant on every connected component of R \ . But the Cantor staircase function is not Lipschitz.
The following proposition by Tapio Rajala 2 (personal communication) connects consequences of permeability and the statement of Corollary 20, that intrinsically Lipschitz continuity for a given exception set implies the overall Lipschitz continuity. It should also be compared to Proposition 13, which states that subsets of permeable sets are permeable.

Proposition 24 Let ⊆ M have the property that every function f : M → Y which is continuous and intrinsically (L-)Lipschitz continuous with exception set is (L-)Lipschitz continuous.
Then every subset 0 of enjoys the same property.
Proof If a function f : M → Y is (L-)Lipschitz continuous with respect to ρ M\ 0 , then, as ρ M\ 0 ≤ ρ M\ , f is (L-)Lipschitz continuous with respect to ρ M\ implying (L-)Lipschitz continuity of f on the whole of M.

Remark 25
With regard to Corollary 20, Theorem 23 and Proposition 13, one may ask the following interesting question: In R d , are the permeable subsets precisely those for which every function is L-Lipschitz iff it is continuous and intrinsically L-Lipschitz with exception set ?
The following example, proposed by Tapio Rajala (personal communication), shows that one may not reduce 'L-Lipschitz' to merely 'Lipschitz' in the above question: Consider the set = ([0, 1]\Q) 2 ⊆ M = [0, 1] 2 . Then the intrinsic metric ρ M\ is given by the 1-distance, see [15,Proposition 3.6]. Hence, if a function is continuous and L-Lipschitz with respect to ρ M\ , then it will be continuous and √ 2L-Lipschitz with respect to the euclidean metric on [0, 1] 2 , and therefore Lipschitz. On the other hand, is not permeable, which we show in the subsequent proposition. Proof Assume that γ is a path from (0, 0) to (1, 1) of length (γ ) < √ 2 + ε. As we can always shorten parts of γ , where its first component is not monotonically increasing, by a vertical line with rational first component, it suffices to assume γ to be given by a functionγ : [0, 1] → [0, 1] of arc length smaller than √ 2 + ε. Similarly, one may replace intervals, whereγ is strictly decreasing, by a horizontal segment with rational function value. So we may assumeγ to be monotonically increasing. As monotonically increasing function, its derivative exists almost everywhere in [0, 1]. The derivative can not be zero almost everywhere, for then the function's arc length were 2 which is not smaller than √ 2 + ε for arbitrary ε, see e.g., [10,Theorem 4]. Thus there must be a set A with λ(A) > 0, whereγ (x) > 0 for all x ∈ A. Since the derivatives are positive on A, it follows thatγ As A is uncountable, Graph(γ | A ) has uncountably many values in the first component, but Graph(γ | A ) intersects the first union of (4) in countably many points. In the same way, the second union intersects Graph(γ | A ) only in countably many points. Therefore Graph(γ | A ) ⊆ Graph(γ ) intersects in uncountably many points. We conclude that is not permeable.
We have shown in Proposition 12 that permeable sets cannot contain interior points. It therefore makes sense to study subsets ⊆ R d that have no interior points relative to R d , and in the next section we further specialize to submanifolds of R d with dimension strictly smaller than d.

Sub-manifolds of R d as Permeable Sets
Definition 27 Let d, m, k ∈ N ∪ {0}, m < d, and let ⊆ R d . We say is an mdimensional C k -submanifold of R d iff for every ξ ∈ there exist open sets U , V ⊆ R d and a C k -diffeomorphism : V → U such that ξ ∈ U and for all y = (y 1 , . . . , y d ) ∈ V it holds (y) ∈ ⇐⇒ y m+1 = · · · = y d = 0. In the case where k = 0, by a C 0 -diffeomorphism we mean a homeomorphism, and we also call a topological submanifold (top-submanifold) .

Corollary 29
The class of Lipschitz submanifolds contains those that possess continuously differentiable mappings , −1 (class C 1 ) as well as those having mappings , −1 that are continuous and piecewise linear, resp. piecewise differentiable, on subdivisions of U , V into polyhedra (class pl, resp. class pdiff).
Before we state the main result of this section, we prove a preparatory lemma. 2 (x + y) and radius 1 2 ( y − x + ε 2 ) 2 − y − x 2 that lies in the hyperplane orthogonal to x − y and passes through 1 2 (x + y). Then the convex hull of B ∪ {x, y}, which we denote by C, is a d-dimensional double cone and, since is a Lebesgue-nullset, we have By Fubini's theorem, From this we conclude that for almost all z ∈ B. We may choose one such z ∈ B, and for this we have 1 ((1 − t)x + tz) + 1 ((1 − t)y + tz) = 0 for almost all t ∈ (0, 1). Thus the proof is finished. Proof Let x, y ∈ R d and ε > 0.
Step 1 Note that, since is a Lipschitz topological submanifold and is thus locally the Lipschitz image of a Lebesgue-nullset, it is itself a Lebesgue-nullset (with respect to R d ) as bi-Lipschitz homeomorphisms pertain measurability of sets and therefore preserve Lebesgue-nullsets, see, e.g., [27,Lemma 7.25]. By virtue of Lemma 30 we may thus restrict our considerations to the case where F := t ∈ [0, 1] : (1−t)x +t y ∈ has Lebesgue measure 0 (in [0, 1]). If F is finite, then we are done. Otherwise assume first that x / ∈ . Since is a closed set, we can find z ∈ such that the line segment γ 1 connecting x and z intersects precisely in z. If we can find a path γ 2 : [a, b] → R d from z to y with (γ 2 ) < y − z + ε and such that γ 2 [a, b] ∩ is finite, then the concatenation γ of the paths γ 1 and γ 2 is a path with the required properties and we are done. Thus we may assume that x ∈ and, by the same argument, that y ∈ .
Step 2 Write g : Since g(t) : t ∈ F ⊆ is compact, there exist finitely many t 1 , . . . , t n ∈ F and bounded open environments U j of g(t j ), V j of 0 and j : V j → U j bi-Lipschitz such that for all z = (z 1 , . . . , z d ) ∈ V j it holds j (z) ∈ ⇐⇒ z m+1 = · · · = z d = 0.
F is a non-empty closed subset of [0, 1], so we can write where the right hand side is a disjoint union and, since λ( and such that γ k ([c k , d k ]) ∩ is finite, then we can construct a path with the required properties. We may therefore concentrate on the case where the whole of g([0, 1]) is contained in a single U j , which we will do in Step 3.
Step 3 Write U := U j , V := V j , := j , L := L j . Since and −1 are Lipschitz continuous with constant L, we have for every finite collection of paths η 1 , . . . , for any finite collection of paths κ 1 , . . . , κ N We repeat the earlier argument to get, for every K ∈ N, a disjoint union of the type (5). Now we choose K big enough to ensure K k=0 (d k − c k ) < ε 2L 2 y−x . We write x k := (1 − c k )x + c k y and y k : t)x + t y that is, κ k is a parametrization of the line-segment from x k to y k . Now κ 0 , . . . , κ K is a finite collection of paths with .
Denote by e d the vector (0, . . . , 0, 1). For every k with k > 0 we construct a path where a ∈ (0, 1) is small enough so that η k (t) ∈ V for all t ∈ [0, 2 k ]. By construction, Now define γ as the concatenation of the following paths: • the paths • η k from x k to y k for k = 0, . . . , K ; • the line segments from y k−1 to x k with lengthsˆ k := x k −y k−1 for k = 1, . . . , K .
Summing up, we get for the length of γ and γ has only finitely many intersections with .

Corollary 32 Let be a Lipschitz submanifold of the R d which is closed as a subset.
Then every continuous function f : R d → R which is intrinsically Lipschitz continuous with exception set is Lipschitz continuous.
Proof This follows immediately from Corollary 20 and Theorem 31. It is readily checked that ρ R 2 = ρ R 2 = ρ R 2 , i.e., both and are permeable. is a sub-manifold of R 2 , which is not a subset of a topologically closed submanifold of the R d , while is closed, but not a submanifold. Note also, that for x = (0, 0), y = (1, 0) there is no path connecting x and y of length smaller than 2 which has a finite intersection with , in contrast to the case of closed Lipschitz submanifolds. Therefore is permeable but not finitely permeable.
The following example shows that one cannot simply dispense with the assumption that the exception set is topologically closed.

Example 34
Recall that the classical Cantor set C is the topological closure of the set C 0 = n k=1 d k 3 −k : n ∈ N, d k ∈ {0, 2} .
Every element from C 0 is the limit of an increasing sequence in R \ C 0 : Denote by D 0 the union of all elements of all these sequences. Then D 0 consists only of isolated points and D 0 ⊇ C 0 = C. Therefore D 0 is a 0-dimensional submanifold of R which is not permeable. By extruding D 0 to D d−1 := D 0 ×R d−1 we get an example of a (d −1)-dimensional C ∞ -submanifold which is not permeable (and not topologically closed).
We conclude this section with more examples of permeable sets. First note that we can somewhat relax the requirement that a Lipschitz manifold be a closed subset of R d , since by Proposition 13 the property ρ R d = ρ R d extends to subsets.
The conclusion of Theorem 31 also holds for unions of closed Lipschitz submanifolds with transversal intersections, with the following notion of transversal intersection: surely. A further hint in this direction is Theorem 1.5 in [3]. It states that for every continuous function g : [0, 1] → R the zeros of B − g have Hausdorff dimension at least 1 2 with positive probability. On the other hand, Theorem 1.2 in [3] states that there exists a function g, which is Hölder continuous with exponent smaller 1 2 such that B − g has isolated zeros with positive probability (nothing is said there about the length of the graph of g). See also the related questions in [24,Open Problem (1)].