Locally Lipschitz functions, cofinal completeness, and UC spaces

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Locally Lipschitz functions, cofinal completeness, and UC spaces Gerald Beer a,∗ , M. Isabel Garrido b,1 a

Department of Mathematics, California State University Los Angeles, 5151 State University Drive, Los Angeles, CA 90032, USA b Instituto de Matemática Interdisciplinar (IMI), Departamento de Geometría y Topología, Universidad Complutense de Madrid, 28040 Madrid, Spain

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 15 October 2014 Available online xxxx Submitted by S. Tikhonov Keywords: Locally Lipschitz function Uniformly locally Lipschitz function Lipschitz in the small function Uniform approximation UC-space Cofinally complete space

Let X, d be a metric space. We find necessary and sufficient conditions on the space for the locally Lipschitz functions to coincide with each of two more restrictive classes of locally Lipschitz functions studied by several authors: the uniformly locally Lipschitz functions and the Lipschitz in the small functions. In the first case, we get the cofinally complete spaces and in the second, the UC spaces. We address this question: to which family of subsets of X, d is the restriction of each function in each class actually Lipschitz? Finally, we determine exactly when the class of uniformly locally Lipschitz functions is uniformly dense in the Cauchy continuous real-valued functions, a class that naturally contains them. In fact, our theorem is valid when the target space is any Banach space. Our density theorem complements the uniform approximation results of Garrido and Jaramillo [12,13]. © 2015 Elsevier Inc. All rights reserved.

1. Introduction Let X, d and Y, ρ be metric spaces with at least two points. A function f : X → Y is called locally Lipschitz if at each x ∈ X, there exists δ > 0 such that the restriction of f to the open ball of radius δ and center x (which we denote by Sd (x, δ)) is Lipschitz: sup

 ρ(f (a), f (b))  : a = b and {a, b} ⊆ Sd (x, δ) < ∞. d(a, b)

Of course the radius δ may be dependent on x: consider f : (0, ∞) → (0, ∞) defined by f (x) = 1/x. If the radius δ can be chosen independent of the point, following [7], we call the function uniformly locally Lipschitz. For example, the squaring function is uniformly locally Lipschitz on R, but for a given * Corresponding author. 1

E-mail addresses: [email protected] (G. Beer), [email protected] (M.I. Garrido). The second author was partially supported by MINECO Project MTM2012-34341 (Spain).

http://dx.doi.org/10.1016/j.jmaa.2015.02.085 0022-247X/© 2015 Elsevier Inc. All rights reserved.

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radius δ, the local Lipschitz constant must be adjusted upward as we move further from the origin. If both the radius and the local Lipschitz constant can be chosen independent of x ∈ X, we call the function Lipschitz in the small. The term originates with Luukkainen [21] and such functions were considered later by Beer, Garrido, and Jaramillo in various combinations [4,7,13]. In analytic terms, this property can be formulated as follows: f is Lipschitz in the small provided there exist δ > 0 and α > 0 such that d(x, w) < δ ⇒ ρ(f (x), f (w)) ≤ αd(x, w). As the locally Lipschitz functions sit within the continuous functions C(X, Y ) from X to Y , the uniformly locally Lipschitz functions sit in the class of (continuous) functions that map Cauchy sequences to Cauchy sequences – the so-called Cauchy continuous functions, while the Lipschitz in the small functions are contained in the larger class of uniformly continuous functions. There are interesting parallels between the three classes, e.g., with respect to their sets of common boundedness. As shown by Beer and Garrido [7], for locally Lipschitz functions (as well as the larger class of continuous functions), these consist of the relatively compact subsets of X; for the class of uniformly locally Lipschitz functions (as well as the Cauchy continuous functions), we get the totally bounded subsets; finally, for the Lipschitz in the small functions (as well as the uniformly continuous functions), we get the Bourbaki bounded subsets. Notice that in each case, the sets of common boundedness form a bornology, i.e., a family of subsets that is hereditary, forms a cover of X, and is stable under finite unions [16]. Other possible parallels seem to be promising: as shown by Garrido and Jaramillo, the locally Lipschitz functions are uniformly dense in C(X, R) [12] while the Lipschitz in the small functions are uniformly dense in the uniformly continuous real-valued functions [13]. Are the uniformly locally Lipschitz functions uniformly dense in the real-valued Cauchy continuous functions? One contribution of this paper is to show that the uniform closure of the real-valued uniformly locally Lipschitz functions need not be the Cauchy continuous functions; in fact, this fails whenever X, d is an infinite-dimensional Banach space. We then characterize those domain spaces for which this kind of uniform approximation can be achieved, and our theorem is valid with R replaced by any Banach space as the target space for our functions. Separately, we provide a short proof of the Garrido–Jaramillo result on the approximation of uniformly continuous real-valued functions by Lipschitz in the small functions that is very different in character than the one they gave [13, Theorem 1]. Perhaps our nicest results are those that give necessary and sufficient conditions on the domain space for pairwise coincidence of the class of locally Lipschitz functions, the class of uniformly locally Lipschitz functions, and the class of Lipschitz in the small functions. Finally, we look at this question: on which subsets of X, d must each locally Lipschitz (resp. uniformly locally Lipschitz, Lipschitz in the small) function be actually Lipschitz? In the first case, we get the relatively compact sets and in the second, we get the totally bounded subsets, but in the third we get a class of sets that can properly contain the Bourbaki bounded sets. While the hoped-for parallelism doesn’t seem to materialize, if we restrict our attention to the subsets on which each Lipschitz in the small functions is both Lipschitz and bounded on the subset, then we do get the Bourbaki bounded subsets and in view of the results of [7], a bona fide parallel to the other two results. Still, there is not systematically a group of parallel results for all three classes of locally Lipschitz functions. As a simple concrete example, we have stability of extensions to the closure for uniformly locally Lipschitz √ functions and for Lipschitz in the small functions but not for locally Lipschitz functions (consider f (x) = x which is locally Lipschitz on the dense subset (0, ∞) of [0, ∞)). Proposition 1.1. Let X, d be a metric space and let A be a dense subset. Let Y, ρ be a second metric space and suppose f ∈ C(X, Y ). If f |A is uniformly locally Lipschitz (resp. Lipschitz in the small), then f is uniformly locally Lipschitz (resp. Lipschitz in the small) as well. Proof. We just prove the first statement. Suppose the restriction of f to each open ball of radius δ in A is Lipschitz. We claim that f is Lipschitz on each open ball of radius 2δ in X. Fix x0 ∈ X and choose a ∈ A

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with d(x0 , a) < 2δ . Choose α > 0 such that if {a1 , a2 } ⊆ A ∩ Sd (a, δ) then ρ(f (a1 ), f (a2 )) ≤ αd(a1 , a2 ). Take {x, w} ⊆ Sd (x0 , 2δ ) and sequences xn  and wn  in A convergent to x and w, respectively. By the triangle inequality, eventually both sequences lie in Sd (a, δ) so that eventually ρ(f (xn ), f (wn )) ≤ αd(xn , wn ). Continuity of f now yields ρ(f (x), f (w)) ≤ αd(x, w) as required. 2 The UC-metric spaces – the metric spaces on which each continuous function is uniformly continuous, and the cofinally complete metric spaces – the metric spaces in which each cofinally Cauchy sequence clusters, play a fundamental role in our subsequent analysis. 2. Preliminaries Let X, d be a metric space. If A ⊆ X, we write cl(A) and A for its closure and set of limit points, respectively. For x ∈ X and A a nonempty subset of X, we put d(x, A) := inf{d(x, a) : a ∈ A}. We also put d(x, ∅) = ∞. If A = ∅, we write diamd (A) for its diameter with respect to the metric d. A subset A of X is called relatively compact provided cl(A) is a compact subset of X. The family of relatively compact subsets forms a bornology on X. A subset A of X is called totally bounded if for each ε > 0 there exists a finite subset F of X such A ⊆ ∪x∈F Sd (x, ε). Each relatively compact subset of X is totally bounded while each totally bounded set is metrically bounded. By an ε-chain of length n from x to y in X, d we mean a finite sequence of points (not necessarily distinct) x0 , x1 , x2 , x3 , . . . , xn in X such that x = x0 , y = xn and for each j ∈ {1, 2, . . . , n}, d(xj−1 , xj ) < ε. We call a subset A of X Bourbaki bounded if for each ε > 0 there is a finite subset F of X and n ∈ N such that each point of A can be joined to some point of F by an ε-chain of length n [7,8,14,15,30]. The Bourbaki bounded subsets lie between the totally bounded subsets and the metrically bounded subsets. All three classes form bornologies. A basic property of metric spaces is their paracompactness, so that each open cover V of X has a continuous partition of unity {pi : i ∈ I} subordinated to it. This means that (i) each pi ∈ C(X, [0, 1]);  (ii) {{x : pi (x) > 0} : i ∈ I} is a locally finite (open) refinement of V; and (iii) ∀x ∈ X, i∈I pi (x) = 1 [31, p. 172]. But one can do better in the metric setting: one can choose the functions to be Lipschitz, as shown by Frolik [11, Theorem 1]. We will exploit this fact in Section 5. If A = ∅, we write Lip(A) for the real-valued Lipschitz functions on A. Two basic references for Cauchy continuous functions are Snipes [28] and Lowen-Colebunders [20]. We content ourselves with listing three well-known facts regarding this class of functions: • each Cauchy continuous function on X, d is continuous; the converse is true if and only if d is a complete metric; • a function is Cauchy continuous on X, d if and only if its restriction to each totally bounded subset is uniformly continuous; • if A is a dense subset of X, d and Y, ρ is a complete metric space and f : A → Y is Cauchy continuous, then f has a (unique) Cauchy continuous extension to X. We now turn our attention to two important classes of metric spaces that have not made there way into standard textbooks in general topology. The UC spaces, first systematically studied by Atsuji [1], is the class of spaces X, d on which each continuous function with values in an arbitrary metric space Y, ρ is uniformly continuous. While the compact metric spaces belong to this class, it is much larger, e.g., it contains N equipped with the usual metric of the real line. These spaces are also called Atsuji spaces [19] and Lebesgue spaces [23,24]. The following result contains a nonexhaustive list of characterizations, all of

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which can be found in the monograph of Beer [3, pp. 54–60] and most in Atsuji’s paper [1]. First, we give a property of sequences that is weaker than the Cauchy property, due to Toader [29]. Definition 2.1. A sequence xn  in a metric space X, d is called pseudo-Cauchy if for each n ∈ N and ε > 0, ∃j > k ≥ n with d(xj , xk ) < ε. In simple words, a sequence is pseudo-Cauchy if pairs of terms are arbitrarily close frequently. We now proceed to our list of defining properties for UC spaces. Theorem 2.2. Let X, d be a metric space. The following conditions are equivalent: (1) X, d is a UC-space; (2) each real-valued continuous function on X is uniformly continuous; (3) each open cover V of X has a Lebesgue number, i.e., ∃λ > 0 such that whenever A ⊆ X with diamd (A) < λ, ∃V ∈ V with A ⊆ V ; (4) whenever A and B are disjoint nonempty closed subsets of X, we have inf {d(a, b) : a ∈ A, b ∈ B} > 0; (5) each sequence xn  in X satisfying limn→∞ d(xn , X\{xn }) = 0 clusters; (6) each pseudo-Cauchy sequence in X with distinct terms clusters; (7) X  is compact, and ∀δ > 0 ∃λ > 0 such that d(x, X  ) > δ ⇒ d(x, X\{x}) > λ. That UC-spaces might be relevant to the present investigation is anticipated by the following result, perhaps first discovered by Beer [2, Theorem 1], and noticed recently and almost concurrently by Di Maio, Meccariello and Naimpally [9] and by Jain and Kundu [19]. Theorem 2.3. Let X, d be a metric space. Then each Cauchy continuous function on X, d with values in an arbitrary metric space is uniformly continuous if and only if the completion of X, d is a UC-space. Turning to cofinal completeness, we need to specify another property of sequences between Cauchyness and pseudo-Cauchyness [5,6,18]. Definition 2.4. A sequence xn  in a metric space X, d is called cofinally Cauchy if for each ε > 0, there exists an infinite subset Nε of N such that whenever {i, j} ⊆ Nε , we have d(xi , xj ) < ε. The metric space is called cofinally complete provided each cofinally Cauchy sequence in X clusters. Evidently, each UC space is cofinally complete, and each cofinally complete space is Cauchy complete. The essential characteristic properties of such spaces are given in our next result. Theorem 2.5. Let X, d be a metric space. The following conditions are equivalent: (1) X, d is a cofinally complete space; (2) each continuous function f on X with values in a second metric space is uniformly locally bounded: there exists μ > 0 such that f restricted to each ball of radius μ is bounded; (3) each continuous real-valued function on X is uniformly locally bounded; (4) for each open cover V of X, there exists μ > 0 such each ball of radius μ in X has a finite subcover; (5) for each open cover V of X, there is an open refinement U and μ > 0 such that for each x ∈ X, Sd (x, μ) intersects at most finitely many members of U; (6) for each locally finite open cover U of X, there exists μ > 0 such that for each x ∈ X, Sd (x, μ) intersects at most finitely many members of U;

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(7) nlc(X) := {x ∈ X : x has no compact neighborhood} is compact, and ∀δ > 0 ∃λ > 0 such that d(x, nlc(X)) > δ ⇒ Sd (x, λ) is relatively compact. Condition (5) is naturally called uniform paracompactness [5,10,17,18,26], and the equivalence of conditions (4) and (5) was proved by Rice [26]. While Rice did not formally note that condition (6) was equivalent to the previous two, he in fact proved it. We will use condition (6) at a key spot here. The equivalence of condition (5) with condition (7) was perhaps first observed by Hohti [17, Theorem 2.1.1]. The equivalence of conditions (1), (2), (3) and (7) is established in [5, Theorem 3.2 and Theorem 3.4]. We rely heavily on condition (3) in the sequel. If we review the conditions of Theorems 2.2 and 2.5, we can easily specify some intriguing conditions that might be added to cofinal completeness to produce a UC-space. Theorem 2.6. Let X, d be a cofinally complete metric space. Each of the following conditions is necessary and sufficient for X, d to be a UC-space: (1) whenever xn  is a sequence in X with limn→∞ d(xn , X\{xn }) = 0, then xn  has a Cauchy subsequence; (2) whenever xn  is a pseudo-Cauchy sequence in X with distinct terms, then xn  has a cofinally Cauchy subsequence; (3) each real-valued uniformly locally bounded continuous function on X is uniformly continuous; (4) whenever V is an open cover of X such that for some μ > 0, diamd (A) < μ ⇒ A has a finite subcover, then V has a Lebesgue number. It is routine to check that condition (1) of Theorem 2.6 is also equivalent to the statement that the completion of X, d is a UC-space; this and many other equivalents are presented in [19]. In the next section, we use this condition to characterize coincidence of the uniformly locally Lipschitz functions with the Lipschitz in the small functions. In passing, we note this following curious equivalence [5, Proposition 3.13]: (a) X, d is totally bounded; (b) each sequence in X, d is cofinally Cauchy; (c) each sequence in X, d is pseudo-Cauchy. 3. Coincidence of classes of locally Lipschitz functions In this section we first characterize those metric spaces on which the class of locally Lipschitz functions and the class of uniformly locally Lipschitz functions coincide, and then those on which the class of uniformly locally Lipschitz functions and the class of Lipschitz in the small functions coincide. These in conjunction immediately tell us when the often larger class of locally Lipschitz functions collapses to the class of Lipschitz in the small functions. As we noted earlier, continuity forces Cauchy continuity precisely when X, d is complete. One might guess that completeness is what is needed for each locally Lipschitz function to be uniformly locally Lipschitz. Instead, something stronger is required. Theorem 3.1. Let X, d be a metric space. The following conditions are equivalent: (1) X, d is a cofinally complete metric space; (2) each locally Lipschitz function on X with values in a metric space Y, d is uniformly locally Lipschitz; (3) each real-valued locally Lipschitz function on X is uniformly locally Lipschitz. Proof. (1) ⇒ (2). Let f : X → Y be locally Lipschitz. Fix y0 ∈ Y ; for each n ∈ N, let Vn be the following open subset of X:

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Vn := {x : ∃ε > 0 ∀a, b ∈ Sd (x,

1 + ε), ρ(f (a), f (b)) < nd(a, b) ∧ f (x) ∈ Sρ (y0 , n)}. n

Clearly, {Vn : n ∈ N} is an open cover of X directed by inclusion. By cofinal completeness/uniform paracompactness, there exists μ > 0 such that each open ball of radius μ has a finite subcover from {Vn : n ∈ N}, and since the cover is directed by inclusion, each such ball is actually contained in some single member of the cover. Fix x ∈ X and choose k ∈ N such that Sd (x, μ) ⊆ Vk . We intend to show that f is Lipschitz on Sd (x, μ). To this end, suppose {a, b} ⊆ Sd (x, μ). If d(a, b) < k1 , then by definition of Vk , we have ρ(f (a), f (b)) < kd(a, b). On the other hand if d(a, b) ≥ k1 , then as {f (a), f (b)} ⊆ Sρ (y0 , k), we have 2k ρ(f (a), f (b)) < = 2k 2 . d(a, b) 1/k We have shown that 2k2 = max{k, 2k 2 } is a Lipschitz constant for f restricted to Sd (x, μ). (2) ⇒ (3). This is obvious. (3) ⇒ (1). Suppose X, d is not cofinally complete. By condition (3) of Theorem 2.5, there exists f ∈ C(X, R) that fails to be uniformly locally bounded, i.e., for each δ > 0 there exists xδ ∈ X such that f restricted to Sd (xδ , δ) fails to be bounded. By the uniform density of the locally Lipschitz functions in C(X, R) [12], there exists a locally Lipschitz function g such that supx∈X |f (x) − g(x)| < 1. Clearly, g must be unbounded on Sd (xδ , δ) for each δ > 0 which makes it impossible for g to be uniformly locally Lipschitz, and we have a contradiction. 2 Theorem 3.2. Let X, d be a metric space. The following conditions are equivalent: (1) each sequence xn  in X with limn→∞ d(xn , X\{xn }) = 0 has a Cauchy subsequence; (2) each uniformly locally Lipschitz function on X with values in an arbitrary metric space Y, d is Lipschitz in the small; (3) each real-valued uniformly locally Lipschitz function on X is Lipschitz in the small. Proof. (1) ⇒ (2). The proof is by contradiction. Assume condition (1) holds and let Y, ρ be a second metric space and let f : X → Y be uniformly locally Lipschitz – say, f is Lipschitz on each open ball of radius δ > 0. If f fails to be Lipschitz in the small, for each n ∈ N, we can find xn , wn in X with 0 < d(xn , wn ) < n1 and ρ(f (xn ), f (wn )) > nd(xn , wn ). By condition (1), xn  has a Cauchy subsequence, so there exists an infinite set of positive integers N1 such that {n, j} ⊆ N1 ⇒ d(xn , xj ) < 2δ . Fix k ∈ N1 ; since Sd (xk , δ) must contain both xn and wn for infinitely many n ∈ N1 , f |Sd (xk ,δ) fails to be Lipschitz, which is impossible. (2) ⇒ (3). This is trivial. (3) ⇒ (1). We prove the contrapositive. Suppose we can find a sequence xn  in X with limn→∞ d(xn , X\{xn }) = 0 that has no Cauchy subsequence. Then {xn : n ∈ N} is not totally bounded [31, p. 182], and by passing to subsequence, we can assume the terms are distinct. By passing again to a subsequence, we can find δ > 0 such that whenever n = j, d(xn , xj ) ≥ δ. As a result, for each x ∈ X, Sd (x, 3δ ) intersects at most one ball from the family {Sd (xn , 3δ ) : n ∈ N}. Define f : X → R by  f (x) =

n− 0

3n δ d(x, xn )

if ∃n with d(x, xn ) < otherwise.

δ 3

Clearly, f is Lipschitz on each ball of radius 3δ . Since limn→∞ d(xn , X\{xn }) = 0, f fails to be Lipschitz in the small. To see this, for all n sufficiently large, we can find wn with

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0 < d(xn , wn ) < d(xn , X\{xn }) +

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δ 1 < , n 3

and so |f (xn ) − f (wn )| = 3n δ d(xn , wn ). Thus, at the same time we can make d(xn , wn ) as small as we please while making |f (xn ) − f (wn )|/d(xn , wn ) as large as we please. 2 Since condition (1) of our last result is equivalent to X, d having UC-completion [19, Theorem 3.8], Theorem 3.2 complements Theorem 2.3 – this, in stark contrast to Theorem 3.1. Theorem 3.3. Let X, d be a metric space. The following conditions are equivalent: (1) X, d is a UC-space; (2) each locally Lipschitz function on X with values in an arbitrary metric space Y, d is Lipschitz in the small; (3) each real-valued locally Lipschitz function on X is Lipschitz in the small. Proof. We need only argue (1) ⇒ (2) and (3) ⇒ (1). If the space X, d has the UC-property, by Theorem 2.6, it is both cofinally complete and each sequence xn  in X with limn→∞ d(xn , X\{xn }) = 0 has a Cauchy subsequence. By the last two theorems, if these occur, then each locally Lipschitz function on X is uniformly locally Lipschitz, and each uniformly locally Lipschitz function on X is Lipschitz in the small – that is, each continuous function on X is uniformly continuous. For (3) ⇒ (1), simply reverse the order of the logical points. 2 4. Subsets on which each member of a class of functions is Lipschitz Let X, d be a metric space and let A be a family of continuous functions on X with values in one or more metric spaces. Minimally, each function in the class must be Lipschitz on each nonempty finite set, and for important classes of functions, there may be no other common subsets of Lipschitzian behavior. We start with the following “folk-theorem”. Theorem 4.1. Let X, d be a metric space and let A be a nonempty subset. The following statements are equivalent: (1) A is a finite set; (2) for each continuous function f from X, d to an arbitrary metric space Y, ρ, f |A is Lipschitz; (3) for each continuous real-valued function f on X, f |A is Lipschitz. Proof. We only sketch the proof of (3) ⇒ (1). If A is infinite, then A is either totally bounded or not. In the first case, we can find a Cauchy sequence in A with distinct terms, say an . Let p be the limit of  of X, d; then the restriction to X of x  p, x  d the sequence in the completion X,  → d( ) defined on the completion is continuous (in fact uniformly continuous) but is not Lipschitz restricted to A. If A is not totally bounded, then A has a uniformly discrete sequence, say an . This means that for some δ > 0, whenever n = j, we have d(an , aj ) > δ, so that in particular the set of terms is closed. Extending an → n · d(an , a1 ) to a continuous function on X produces a function that fails to be Lipschitz on A. 2 The restriction of each real-valued uniformly continuous function on the real line to each uniformly discrete subset is easily seen to be Lipschitz, so that continuity cannot be replaced by uniform continuity in condition (3) above (but see [22, Theorem 3]).

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We next turn to the class of locally Lipschitz functions, where our next result is also in some sense known; for example, the equivalence of conditions (1) and (3) is essentially due to Scanlon [27, Theorem 1] (see also [25, p. 17]). But for completeness and since the proof is short, as a courtesy to the reader, we include it. Theorem 4.2. Let X, d be a metric space and let A be a nonempty subset. The following statements are equivalent: (1) A is relatively compact; (2) for each locally Lipschitz function f from X, d to an arbitrary metric space Y, ρ, f |A is Lipschitz; (3) for each real-valued locally Lipschitz function f on X, f |A is Lipschitz. Proof. For (1) ⇒ (2), suppose A has compact closure and f is locally Lipschitz, yet f |A fails to be Lipschitz. Then we can find distinct xn , wn in A such that ρ(f (xn ), f (wn )) >n d(xn , wn )

(n ∈ N).

By passing to subsequences, we can assume that xn  converges to a ∈ cl(A) while wn  converges to b ∈ cl(A). Clearly, a and b are distinct, else the locally Lipschitz condition on f is violated. But if a = b, then there exists δ > 0 such that for all n sufficiently large, d(xn , wn ) > δ. This forces supn∈N ρ(f (xn ), f (wn )) = ∞, violating the boundedness of f on A as guaranteed by continuity and the compactness of cl(A). The implication (2) ⇒ (3) is trivial. Finally, for (3) ⇒ (1), suppose A is not relatively compact, and let an  be a sequence of distinct terms in A that has no cluster point in X. For each n ≥ 2, put δn := min{ n1 , 13 d(an , {aj : j = n})}, and let fn : Sd (an , δn ) → R be a Lipschitz function mapping an to nd(a1 , an ) and which is zero on {x : 12 δn ≤ d(x, an ) < δn }. Finally, put  f (x) :=

fn (x) if ∃n ≥ 2 with d(x, an ) < δn 0 otherwise.

Since limn→∞ δn = 0 and an  has no cluster point, the family {Sd (an , δn ) : n ≥ 2} must be locally finite. From this, it is easy to see that f is locally Lipschitz, for on each ball that hits at most finitely many members of the family, f can be realized as a maximum of finitely many Lipschitz functions. On the other hand, for each n ≥ 2, |f (an ) − f (a1 )|/d(an , a1 ) = n so that f is not Lipschitz on A. 2 Moving on to the class of uniformly locally Lipschitz functions, the next result is new. Theorem 4.3. Let X, d be a metric space and let A be a nonempty subset. The following statements are equivalent: (1) A is totally bounded; (2) for each uniformly locally Lipschitz function f from X, d to an arbitrary metric space Y, ρ, f |A is Lipschitz; (3) for each real-valued uniformly locally Lipschitz function on X, f |A is Lipschitz. Proof. (1) ⇒ (2). Choose δ > 0 such that for each x ∈ X, f is Lipschitz on the ball Sd (x, δ). By the definition of total boundedness, there is a finite subset F of X such that A ⊆ ∪x∈F Sd (x, 2δ ). For each x ∈ F , let αx be a Lipschitz constant for the restriction of f to Sd (x, δ) and then put α := max{αx : x ∈ F }. By the Cauchy continuity of f alone, μ := diamρ f (A) is finite [7,28]. Let a1 and a2 be distinct points of A. If d(a1 , a2 ) < 2δ , then for some x ∈ F , we have {a1 , a2 } ⊆ Sd (x, δ) from which we obtain ρ(f (a1 ), f (a2 )) ≤ αd(a1 , a2 ). On the other hand, if d(a1 , a2 ) ≥ 2δ , we obtain

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G. Beer, M.I. Garrido / J. Math. Anal. Appl. ••• (••••) •••–•••

9

2μ ρ(f (a1 ), f (a2 )) ≤ . d(a1 , a2 ) δ These estimates show that a Lipschitz constant for f |A is max {α, 2μ δ }. The implication (2) ⇒ (3) is obviously true. For (3) ⇒ (1), suppose A is not totally bounded. For some δ > 0, there is a sequence of distinct terms an  in A such that d(ak , an ) ≥ δ whenever k = n. By construction, for each x ∈ X, Sd (x, 3δ ) can intersect at most one ball in the family {Sd (an , 3δ ) : n ∈ N}. For each n ≥ 2, let fn : Sd (an , 3δ ) → R be defined by δ 3n fn (x) = ( − d(x, an )) d(a1 , an ). 3 δ Then fn is Lipschitz on Sd (an , 3δ ) and fn (an ) = nd(a1 , an ). It follows that  f (x) :=

fn (x) 0

if ∃n ≥ 2 with d(x, an ) < otherwise

δ 3

is Lipschitz on each open ball of radius 3δ as f can be realized as the maximum of the zero function and at most one other Lipschitz function on the ball. But as f (a1 ) = 0, we obtain |f (a1 ) − f (an )| = nd(a1 , an ) for each n ≥ 2, so that f ∈ / Lip(A). 2 With the parallel results for common sets of boundedness of locally Lipschitz functions, uniformly locally Lipschitz functions, and Lipschitz in the small functions as established in [7] in mind, we might anticipate that each globally defined Lipschitz in the small function will be Lipschitz on a subset A precisely when A is Bourbaki bounded. It turns out that we usually get a properly larger family of subsets; in fact, such sets can fail to be metrically bounded. For example, it is easily seen that a globally defined Lipschitz in the small function is Lipschitz when restricted to a nonempty subset A of the metric space X, d such that d|A×A is an almost convex metric (see, e.g., [3]): whenever a1 = a2 in A and α satisfies d(a1 , a2 ) < α, then for each β ∈ (0, α), there exists a3 ∈ A with d(a1 , a3 ) < β and d(a2 , a3 ) < α − β. Thus, in particular, if X, d is isometrically embedded in a normed linear space, then on each convex subset of X relative to the embedding, each globally defined Lipschitz in the small function is Lipschitz. Garrido and Jaramillo [13] called a metric space X, d small determined if each Lipschitz in the small function f on X is already Lipschitz on X. Thus it is natural to call a subset A of a metric space X, d small determined if for each globally defined Lipschitz in the small function f , f |A is Lipschitz. Our next result characterizes the Bourbaki bounded subsets among the small determined subsets. Theorem 4.4. Let A be a nonempty subset of a metric space X, d. Then A is Bourbaki bounded if and only if it is both d-bounded and small determined. Proof. For necessity, it is clear that each Bourbaki bounded set is metrically bounded. Let Y, ρ be a second metric space and suppose f : X → Y is Lipschitz in the small. Choose δ > 0 and α > 0 such that whenever d(x1 , x2 ) < δ, then ρ(f (x1 ), f (x2 )) ≤ αd(x1 , x2 ). In particular, this holds when {x1 , x2 } ⊆ A. By the uniform continuity of f alone, f |A is bounded [1,7], so with μ = diamρ f (A), we see that if {a1 , a2 } ⊆ A and d(a1 , a2 ) ≥ δ, then ρ(f (a1 ), f (a2 )) ≤ μδ d(a1 , a2 ). In view of these estimates, the restriction of f to A is Lipschitz. For sufficiency, suppose A = ∅ is d-bounded and each Lipschitz in the small function on X is Lipschitz restricted to A. Thus, each Lipschitz in the small function on X is bounded when restricted to A. By [7, Theorem 3.3], A is Bourbaki bounded. 2

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Doctopic: Real Analysis

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G. Beer, M.I. Garrido / J. Math. Anal. Appl. ••• (••••) •••–•••

Theorem 4.4 now yields a result that parallels Theorems 4.2 and 4.3, for if each locally (resp. uniformly locally) Lipschitz function is Lipschitz restricted to a nonempty subset A, then one gets the boundedness of each such function on A for free. Theorem 4.5. Let A be a nonempty subset of a metric space X, d. The following statements are equivalent: (1) A is Bourbaki bounded; (2) for each Lipschitz in the small function f from X, d to an arbitrary metric space Y, ρ, f |A is both Lipschitz and bounded; (3) for each real-valued Lipschitz in the small function f on X, f |A is both Lipschitz and bounded. Proof. (1) ⇒ (2) follows from Theorem 4.4 and [7, Theorem 3.3]. The implication (2) ⇒ (3) is trivial. Finally, (3) ⇒ (1) follows simply from the boundedness of each real-valued Lipschitz in the small function restricted to A. 2 At this time, a direct characterization of the family of small determined subsets of a metric space themselves eludes us. We leave this as an open question to the interested reader. The small determined subsets of a metric space, while a hereditary family, are not in general stable under finite unions [13]. For example, consider A = {(0, β) : β ∈ R} and B = {(1, β) : β ∈ R} and let our metric space X be A ∪ B equipped with the Euclidean metric of the plane. While A and B being convex subsets are small determined, A ∪ B fails to be. Let f : X → R be the zero function on A and let f map (1, β) to β. As |f (x1 ) − f (x2 )| ≤ 1 · d(x1 , x2 ) whenever d(x1 , x2 ) ≤ 12 , f is Lipschitz in the small, while f ∈ / Lip(X) because for β > 0, |f (0, β) − f (1, β)| = β = β · d((0, β), (1, β)). In contrast, Theorem 4.4 shows that the bounded small determined subsets do form a bornology. One should also notice that the property of being small determined for a subset A of a metric space can depend on the metric space in which the subset is isometrically embedded. For example, A ∪ B as just described is small determined considered as a subset of R2 where every subset is small determined. 5. Density of the uniformly locally Lipschitz functions in the Cauchy continuous functions As we noted earlier, the real-valued locally Lipschitz functions on an arbitrary metric space X, d are uniformly dense in the continuous functions, while the real-valued Lipschitz in the small functions are uniformly dense in the uniformly continuous functions [12,13]. Given the parallelism between our three classes of locally Lipschitz functions, one might guess that the uniformly locally Lipschitz functions are in general uniformly dense in the Cauchy continuous functions, but this is not to be. We first provide a concrete counterexample, and then proceed to show that this occurs exactly when our metric space has a cofinally complete completion. For sufficiency, our main tool is Frolik’s theorem on the existence of Lipschitz partitions of unity subordinated to any open cover of an arbitrary metric space [11, Theorem 1]. Using this approach, we may replace the target space R by any Banach space. Similarly, it can be shown using Lipschitz partitions of unity that the uniform density of the locally Lipschitz functions in C(X, R) holds when the target space is an arbitrary normed linear space; Garrido and Jaramillo [12] obtain this result for real-valued functions using very different tools. Example 5.1. We show that for any infinite dimensional Banach space X, || · ||, the uniformly locally Lipschitz real-valued functions are not uniformly dense in the Cauchy continuous functions. Let x ∈ X be a norm-one vector, and for each n ∈ N, let xn = n2 x. Let Bn be the closed ball with radius n1 with center xn . By infinite dimensionality, the ball is noncompact and so we can find an unbounded fn ∈ C(Bn , R). Now ∪∞ n=1 Bn is a closed set, so by the Tietze extension theorem [31, p. 103], there exists f ∈ C(X, R) such that for each n ∈ N, f |Bn = fn . Since the metric of the norm is complete, f is Cauchy continuous.

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Doctopic: Real Analysis

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G. Beer, M.I. Garrido / J. Math. Anal. Appl. ••• (••••) •••–•••

11

Now let g : X → R be an arbitrary uniformly locally Lipschitz function. By definition, ∃δ > 0, ∀x ∈ X, g restricted to Sd (x, δ) is Lipschitz and is therefore bounded. Choose n > 1δ ; we compute supx∈X |f (x) − g(x)| ≥ supx∈Bn |f (x) − g(x)| = ∞, and this shows that f cannot be uniformly approximated by uniformly locally Lipschitz functions. Before we come to our main result of this section, we need this simple lemma. ˆ be its completion. Suppose Y, ρ is a complete metric ˆ d Lemma 5.2. Let X, d be a metric space and let X, space. Then the uniformly locally Lipschitz functions on X are uniformly dense in the Cauchy continuous ˆ are uniformly functions on X with values in Y if and only if the uniformly locally Lipschitz functions on X ˆ dense in C(X, Y ). ˆ Y ); Proof. Suppose we have uniform density in the Cauchy continuous functions on X. Let fˆ ∈ C(X, then fˆ is Cauchy continuous so that its restriction f to X is Cauchy continuous as well. Let ε > 0 and choose g : X → Y that is uniformly locally Lipschitz and supx∈X ρ(f (x), g(x)) ≤ ε. Since uniformly locally ˆ → Y which Lipschitz functions are Cauchy continuous and ρ is complete, there is a unique extension gˆ : X ˆ by Proposition 1.1 must be uniformly locally Lipschitz. It is routine to verify that supxˆ∈Xˆ ρ(f (ˆ x), gˆ(ˆ x)) ≤ ε. ˆ Y ). If f : X → Y is Cauchy continuous, by the Conversely, suppose we have uniform density in C(X, ˆ Since fˆ is continuous (without completeness of ρ, we can extend f to a Cauchy continuous function fˆ on X. ˆ considering the completeness of d), we can find a sequence ˆ gn  of uniformly locally Lipschitz functions on ˆ whence their restrictions to X converge uniformly to f . 2 ˆ uniformly convergent to f, X Theorem 5.3. Let X, d be a metric space. The following conditions are equivalent: ˆ of X is cofinally complete; ˆ d (1) the completion X, (2) whenever Y, || · || is a Banach space and f : X → Y is Cauchy continuous, then f can be uniformly approximated by uniformly locally Lipschitz functions; (3) whenever f : X → R is Cauchy continuous, then f can be uniformly approximated by uniformly locally Lipschitz functions. ˆ f has a unique Cauchy Proof. (1) ⇒ (2). Since Y is a complete metric space, and X is dense in X, ˆ choose δxˆ > 0 ˆ → Y . Of course, fˆ is continuous, so given ε > 0, for each x continuous extension fˆ : X ˆ∈X ˆ ˆ such d(ˆ x, w) ˆ < δxˆ ⇒ ||f (ˆ x) − f (w)|| ˆ < ε. For each x ˆ, put Vxˆ := Sdˆ(ˆ x, δxˆ ) and let {pi : i ∈ I} be a Lipschitz ˆ whose existence is guaranteed by Frolik’s theorem [11]. For partition of unity subordinated to {Vxˆ : x ˆ ∈ X} each i ∈ I, set Ui = {ˆ x : pi (ˆ x) > 0} and choose x ˆ(i) with Ui ⊆ Vxˆ(i) . Of course, {Ui : i ∈ I} is a locally ˆ finite open refinement of {Vxˆ : x ˆ ∈ X}, so by condition (6) of Theorem 2.5, we can find β > 0 such that ˆ hits at most finite many Ui . Since the Lipschitz functions defined on a given each open ball of radius β in X ˆ with values in Y form a vector space, by local finiteness gˆ :=  fˆ(ˆ subset of X x(i))pi (·) is well-defined i∈I ˆ and is Lipschitz on each ball of radius β. It remains to show that supx∈X ||f (x) − gˆ(x)|| ≤ ε because the restriction of gˆ to X will be uniformly locally Lipschitz as well. To this end, fix x ∈ X and let Ui1 , Ui2 , . . . , Uin have nonempty intersection with Sdˆ(x, β). Let ˆ x Ui1 , Ui2 , . . . , Uim with 1 ≤ m ≤ n be those Ui in which x lies. For each j ≤ m, d(x, ˆ(ij )) < δxˆ(ij ) , so that pij (x)||fˆ(x) − fˆ(ˆ x(ij ))|| < pij (x) · ε for j ≤ m. In addition, for all j with m < j ≤ n, we have pij (x)||fˆ(x) − fˆ(ˆ x(ij ))|| = pij (x) · ε because pij (x) = 0. We compute ||ˆ g (x) − fˆ(x))|| = ||

n  j=1

fˆ(ˆ x(ij ))pij (x) − fˆ(x)pij (x)|| ≤

n  j=1

pij (x) · ε = ε.

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Doctopic: Real Analysis

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G. Beer, M.I. Garrido / J. Math. Anal. Appl. ••• (••••) •••–•••

(2) ⇒ (3). This is trivial. ˆ is not cofinally complete. By condition (3) of Theorem 2.5 we can find fˆ ∈ C(X, ˆ R) (3) ⇒ (1). Suppose X that is not uniformly locally bounded. Arguing just as in the proof of Theorem 3.1, we see that fˆ cannot ˆ Apply Lemma 5.2. 2 be uniformly approximated by uniformly locally Lipschitz functions on X. The reader should notice that our uniformly locally Lipschitz approximates of a given Cauchy continuous function with values in a Banach space Y as described in (1) ⇒ (2) map each ball of a prescribed radius in the domain into a polytope in the Banach space. In particular, each such ball is mapped to a relatively compact subset of Y . Corollary 5.4. Let X, d be a complete metric space. Then X is cofinally complete if and only if each continuous function on X with values in a Banach space can be uniformly approximated by uniformly locally Lipschitz functions. 6. A short proof of the Garrido–Jaramillo theorem on the approximation of uniformly continuous functions Garrido and Jaramillo [13, Theorem 1] proved that each uniformly continuous real-valued function on a metric space X, d can be uniformly approximated by Lipschitz in the small functions by taking a uniform cover approach. The alternative proof that follows involves a natural modification of a standard Lipschitz regularization technique frequently used in convex and variational analysis (see, e.g., [3]). It is motivated by the following consideration: f : X → R is Lipschitz in the small if and only if for some δ > 0 and α > 0, whenever (x, γ) lies in the epigraph of f , {(w, β) : d(x, w) < δ and β ≥ γ + αd(x, w)} is contained in the epigraph of f . Theorem 6.1. Let X, d be a metric space and let f : X → R be a uniformly continuous function. Given ε > 0, there exists a Lipschitz in the small function g : X → R with supx∈X |f (x) − g(x)| ≤ ε. Proof. Choose δ > 0 such that for x, w ∈ X, d(x, w) < δ ⇒ |f (x) − f (w)| < ε. Choose k ∈ N such that (♠) kδ 2 > 1 + ε. Define g : X → R by g(x) := infw∈Sd (x,δ) f (w) + kd(x, w). Note that for each x ∈ X, f (x) ≥ g(x) ≥ infw∈Sd (x,δ) f (w) ≥ f (x) − ε. It remains to show that g is Lipschitz in the small. We will show that if d(x1 , x2 ) < 2δ , then |g(x2 )−g(x1 )| ≤ kd(x2 , x1 ). By symmetry, it suffices to show that g(x2 ) − g(x1 ) ≤ kd(x2 , x1 ). To this end, let λ ∈ (0, 1) be arbitrary, and choose w1 ∈ Sd (x1 , δ) such that f (w1 ) + kd(x1 , w1 ) < g(x1 ) + λ. Now either d(x1 , w1 ) ≥ 2δ or d(x1 , w1 ) < 2δ . In the first case, we compute using (♠) f (w1 ) + kd(x1 , w1 ) > f (x1 ) − ε + 1 + ε = f (x1 ) + 1 > g(x1 ) + λ, which is impossible. We are left with d(x1 , w1 ) < estimate g(x2 ), we have

δ 2

so that d(x2 , w1 ) < δ as well. As w1 can be used to

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Doctopic: Real Analysis

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G. Beer, M.I. Garrido / J. Math. Anal. Appl. ••• (••••) •••–•••

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g(x2 ) ≤ f (w1 ) + kd(x2 , w1 ) = f (w1 ) + kd(x1 , w1 ) + kd(x2 , w1 ) − kd(x1 , w1 ) < g(x1 ) + λ + kd(x2 , x1 ). Since λ ∈ (0, 1) was arbitrary, the proof is complete. 2 References [1] M. Atsuji, Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8 (1958) 11–16. [2] G. Beer, More about metric spaces on which continuous functions are uniformly continuous, Bull. Aust. Math. Soc. 33 (1986) 397–406. [3] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers, Dordrecht, Holland, 1993. [4] G. Beer, On metric boundedness structures, Set-Valued Anal. 7 (1999) 195–208. [5] G. Beer, Between compactness and completeness, Topology Appl. 155 (2008) 503–514. [6] G. Beer, G. Di Maio, The bornology of cofinally complete subsets, Acta Math. Hungar. 134 (2012) 322–343. [7] G. Beer, M.I. Garrido, Bornologies and locally Lipschitz functions, Bull. Aust. Math. Soc. 90 (2014) 257–263. [8] N. Bourbaki, Elements of Mathematics, General Topology, Part 1, Hermann, Paris, 1966. [9] G. Di Maio, E. Meccariello, S. Naimpally, Decompositions of UC spaces, Questions Answers Gen. Topology 22 (2004) 13–22. [10] J. Fried, Z. Frolik, A characterization of uniform paracompactness, Proc. Amer. Math. Soc. 89 (1983) 537–540. [11] Z. Frolik, Existence of ∞ partitions of unity, Rend. Semin. Mat. Univ. Politec. Torino 42 (1984) 9–14. [12] M.I. Garrido, J. Jaramillo, Homomorphisms on function lattices, Monatsh. Math. 141 (2004) 127–146. [13] M.I. Garrido, J. Jaramillo, Lipschitz-type functions on metric spaces, J. Math. Anal. Appl. 340 (2008) 282–290. [14] M.I. Garrido, A.S. Meroño, New types of completeness in metric spaces, Ann. Acad. Sci. Fenn. Math. 39 (2014) 733–758. [15] J. Hejcman, Boundedness in uniform spaces and topological groups, Czechoslovak Math. J. 9 (1959) 544–563. [16] H. Hogbe-Nlend, Bornologies and Functional Analysis, North-Holland, Amsterdam, 1977. [17] A. Hohti, On uniform paracompactness, in: Ann. Acad. Sci. Fenn. Math. Diss., vol. 36, 1981, pp. 1–46. [18] N. Howes, Modern Analysis and Topology, Springer-Verlag, New York, 1995. [19] T. Jain, S. Kundu, Atsuji completions: equivalent characterizations, Topology Appl. 154 (2007) 28–38. [20] E. Lowen-Colebunders, Function Classes of Cauchy Continuous Functions, Marcel Dekker, New York, 1989. [21] J. Luukkainen, Rings of functions in Lipschitz topology, Ann. Acad. Sci. Fenn. Math. 4 (1978/1979) 119–135. [22] G. Marino, When is any continuous function Lipschitzian?, Extracta Math. 13 (1998) 107–110. [23] A. Monteiro, M. Peixoto, Le nombre de Lebesgue et la continuité uniforme, Port. Math. 10 (1951) 105–113. [24] S. Nadler, T. West, A note on Lebesgue spaces, Topology Proc. 6 (1981) 363–369. [25] W. Pfeffer, The Divergence Theorem and Sets of Finite Perimeter, CRC Press, Boca Raton, FL, 2012. [26] M. Rice, A note on uniform paracompactness, Proc. Amer. Math. Soc. 62 (1977) 359–362. [27] C. Scanlon, Rings of functions with certain Lipschitz properties, Pacific J. Math. 32 (1970) 197–201. [28] R. Snipes, Functions that preserve Cauchy sequences, Nieuw Arch. Wiskd. 25 (1977) 409–422. [29] G. Toader, On a problem of Nagata, Mathematica 20 (1978) 78–79. [30] T. Vroegrijk, Uniformizable and realcompact bornological universes, Appl. Gen. Topol. 10 (2009) 277–287. [31] S. Willard, General Topology, Addison–Wesley, Reading, MA, 1970.