SPLIT CONTINUITY 1. Introduction

Home , Net (mathematics), Tietze extension theorem

SPLIT CONTINUITY

GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON

Abstract. In the context of functions between metric spaces, we introduce a class of functions that simultaneously provides characteristic answers to the following two questions: (1) What property must a function f between metric spaces X and Y have so that when it is followed by an arbitrary continuous real-valued function, the composition is either upper semicontinuous or lower semicontinuous at each point of X? (2) What property must a function f between metric spaces X and Y have so that there exists a second function g from X to Y such that the multifunction x 7→ {f(x), g(x)} is globally upper semicontinuous? Particular attention is given to the classical Thomae function and arbitrary characteristic functions as members of this class.

1. Introduction Let hX, di and hY, ρi be metric spaces. By a multifunction from X to Y , we mean a function Γ with domain X that assigns to each x ∈ X a nonempty subset Γ(x) of Y . Of course, a single-valued function f : X → Y is naturally associated with the multifunction x 7→ {f(x)}. One natural way to deﬁne global continuity for Γ that reduces to ordinary global continuity when the values of Γ are singletons is to insist that for each open subset V of Y, {x ∈ X : Γ(x) ⊆ V } is open in X. When this requirement is met, we say that Γ is globally upper semicontinuous. Now suppose f and g are two (single-valued) functions from X to Y ; it is natural to inquire when x 7→ {f(x), g(x)} is globally upper semicontinuous. For this to occur, both and f and g must satisfy a property that we call global split continuity. While it is not true that global split continuity of both f and g ensures global upper semicontinuity of x 7→ {f(x), g(x)}, for each globally split continuous function f : X → Y , there is a naturally associated second globally split continuous function f ∗ such that the multifunction x 7→ {f(x), f ∗(x)} is globally upper semicontinuous. One question to which we devote attention is this: given a globally split continuous function f, when does there exist a second globally split continuous function g such that g∗ = f? While the class of globally split continuous functions includes the continuous functions, it also includes classical functions that are presented to students of el- ementary analysis as somehow ill-behaved, e.g, the Thomae function. Each characteristic function is split continuous, and so as long as X has a non-Borel subset,

Date: January 3, 2019. 2010 Mathematics Subject Classification. Primary 54E40,54C30,54C60 Secondary 26A15, 26A30. Key words and phrases. split continuous function, semicontinuous real-valued function, upper semicontinuous multifunction, characteristic function, oscillation function, Thomae function. The work of the second and third authors was partially supported by the National Science Foundation [grant number DMS-1600778]. 1 2 GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON there are non-Borel split continuous functions. Thus, some bad objects become good ones from our perspective. For real-valued functions, global split continuity is closely related to upper and lower semicontuity as defined in a standard course in analysis. First, a globally split continuous real-valued function is either upper semicontinuous or lower semicontinuous at each point of the domain - the type of semicontinuity can vary with the point. For instance, in the case of the Dirichlet function, we have upper semicontinuity at each rational number and lower semicontinuity at each irrational number, while for the Thomae function (considered as a function on [0,1]), we have upper semicontinuity at each rational and continuity at each irrational. Second, global split continuity of a function f : X → Y is necessary and sufficient for g ◦ f to be semicontinuous at each point of X for each continuous real-valued function g with domain Y . We show that the upper semicontinuous oscillation function for each globally split continuous function is also globally split continuous. Finally, for a real-valued function f whose upper and lower envelopes are both real-valued, split continuity is both necessary and sufficient for the closure of the graph of f to be expressible as the union of the graphs of its upper and lower envelopes.

2. Preliminaries We denote the positive integers, the integers, the rationals and the reals by N, Z, Q, and R, respectively. In the sequel, we assume that all metric spaces include at least two points. If p is a point of the metric space hX, di, we denote the open ball with center p and radius α > 0 by Bd(p, α). If A is a subset of X, we denote its complement, closure, interior, boundary and set of limits points by Ac, cl(A), int(A), bd(A) and A0, respectively. By diamd(A), we mean the usual diameter of a nonempty subset A, and for x ∈ X, we put d(x, A) := inf{d(x, a): a ∈ A}. We say A is nowhere dense if int(cl(A)) = ∅. The family of nowhere dense sets is stable under ﬁnite unions and under taking subsets of its members [14, p. 2]; the family contains the singletons if and only if X0 = X. By a regular open set V , we mean an open set such that int(cl(V )) = V [14, p. 20]; dually, a closed set F is called regular if F = cl(int(F )). We call a nonempty subset E of hX, di ε-discrete provided whenever e1 and e2 are distinct members of E, we have d(e1, e2) ≥ ε. Using Zorn’s lemma, it is easy to show that each ε-discrete subset E of A is contained in a maximal ε-discrete subset C of A, in which case ∀a ∈ A, ∃c ∈ C with d(c, a) < ε. We call a maximal ε-discrete subset of A an ε-net for A. Given a function f : hX, di → hY, ρi, we denote its graph by gr(f). If A is a nonempty subset of X, we denote the restriction of f to A by f A. If A ⊆ X, we denote its characteristic function by χA. We write C(X,Y ) for the space of continuous functions from X to Y . For a metric space hX, di, we say that a function f : X → (−∞, ∞] is upper semicontinuous (u.s.c.) at p ∈ X if whenever hxni converges to p, we have f(p) ≥ lim supn→∞ f(xn). For a real-valued function f, (i) upper semicontinuity at p means that for each ε > 0 there exists δ > 0 such that d(x, p) < δ implies f(x) < f(p) + ε, and (ii) we have global upper semicontinuity if and only if the inverse image of each open ray of the form (−∞, α) is open. For each real-valued function SPLIT CONTINUITY 3 f on X, there is a smallest globally upper semicontinuous function f on X that majorizes f, called the upper envelope of f, and deﬁned by

f(x) := infε>0 sup{f(w): d(w, x) < ε} (x ∈ X).

The following facts about the upper envelope of a real-valued function f are all well-known:

• for each p ∈ X there exists a sequence hxni convergent to p for which f(p) = limn→∞ f(xn), and this is the largest limit that can be so realized; • f(p) is ﬁnite if and only if f is locally bounded above at p; • f(p) = f(p) if and only if f is upper semicontinuous at p.

Dually, we say f : X → [−∞, ∞) is lower semicontinuous (l.s.c.) at p ∈ X if whenever hxni converges to p, we have f(p) ≤ lim infn→∞ f(xn). If f is real-valued, then f is continuous at p if and only if f is both u.s.c. and l.s.c. at p. Given a real- valued function f on X, there is a largest globally lower semicontinuous function f on X that is majorized by f, called the lower envelope of f, and deﬁned by

f(x) := supε>0 inf{f(w): d(w, x) < ε} (x ∈ X).

It is routine to show that at each p ∈ X, f(p) ≤ f(p) with equality occuring if and only if f is continuous at p. The reader is invited to dualize the three bullet- points above for the lower envelope of a real-valued function. With the second bullet-point and its dual in mind, we can say that both f(p) and f(p) are finite if and only if f is locally bounded at p. For further information about semicontinuous extended real-valued functions defined on topological spaces, we refer the reader to [4, 10]; the first monograph identifies upper (resp. lower) semicontinuous functions with their hypographs (resp. epigraphs) in X × R, while the second offers characterizations of various classes of topological spaces in terms of the behavior of the real-valued semicontinuous functions defined on them, e.g., by so-called sandwich theorems [10, p. 88]. If Γ is a multifunction from hX, di to hY, ρi, we express this rule of assignment by the notation Γ : X ⇒ Y . The graph of Γ is of course given by

gr(Γ) := {(x, y): x ∈ X, y ∈ Γ(x)}.

If Γ0 is a second multifunction from X to Y , we call Γ0 a proper submultifunction of Γ if gr(Γ0) is a proper subset of gr(Γ). We say that Γ is upper semicontinuous at p ∈ X if whenever V is a neighborhood of Γ(p), there exists δ > 0 such that d(x, p) < δ implies that Γ(x) ⊆ V . Global upper semicontinuity of Γ as we deﬁned it in the introduction amounts to upper semicontinuity at each point of the domain. It is easy to show that a globally upper semicontinuous multifunction with closed values also has closed graph. If f : X → R, then f is upper semicontinuous at p ∈ X as a single-valued function if and only if the multifunction x 7→ (−∞, f(x)] is upper semicontinuous at p. Globally upper semicontinuous multifunctions with compact values are usually called usco maps in the literature. Other basic facts about upper semicontinuity of 4 GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON multifunctions and the companion notion of lower semicontinuity can be found in the monographs [2, 4, 7, 13]. We now come to the fundamental concept to be introduced in this article.

Deﬁnition 2.1. Let hX, di and hY, ρi be metric spaces. Let f : X → Y be a function and let p ∈ X. We say that f is split continuous at p if ∃y ∈ Y such that

(1) ∀ε > 0, f(Bd(p, ε)) ∩ Bρ(y, ε) 6= ∅;

(2) ∀ε > 0, ∃δ > 0 such that f(Bd(p, δ)) ⊆ Bρ(f(p), ε) ∪ Bρ(y, ε).

In Deﬁnition 2.1, we could replace condition (1) by this formally stronger but equivalent one:

∀ε > 0, ∀δ > 0, f(Bd(p, δ)) ∩ Bρ(y, ε) 6= ∅.

While our definition does not say the conditions cannot be satisfied by more than one y ∈ Y , this is indeed the case. First, if f is continuous at p, then condition (1) can only hold for y = f(p), and condition (2) holds for y = f(p). Conversely, if the conditions are satisfied with respect to y = f(p), then condition (2) alone forces continuity of f at p. Let us move to the case that the conditions are satisfied by some y 6= f(p). Suppose in addition that there exists y1 satisfying the above properties where y1 6= 1 f(p) and y1 6= y. Let ε = 2 min{ρ(y, y1), ρ(f(p), y1)}. Then there exists δ ∈ (0, ε) such that f(Bd(p, δ)) ⊆ Bρ(f(p), ε)∪Bρ(y, ε). As a result, f(Bd(p, δ))∩Bρ(y1, δ) = ∅, a contradiction. In what follows, if f is split continuous at p ∈ X, we let yp ∈ Y be the unique y-value which satisfies the above definition. If yp 6= f(p), i.e., if f is not continuous at p, we say that f is strictly split continuous at p. The purpose of the following proposition is to state two obvious sequential properties that a function f that is split continuous at a point p must have. Taken together, they are not sufficient for split continuity of f at p: consider f : R → R defined by f(x) = 1 if x ≤ 0 and f(x) = 1/x if x > 0 where p = 0 and yp = 1. Proposition 2.2. Let hX, di and hY, ρi be metric spaces. Suppose that the function f : X → Y is split continuous at p ∈ X.

(1) There is a sequence hxni convergent to p for which hf(xn)i → yp;

(2) Whenever hxni → p and hf(xn)i → y ∈ Y , then either y = f(p) or y = yp.

It is useful to sequentially characterize strict split continuity at a point. Proposition 2.3. The following are equivalent for a function f : hX, di → hY, ρi.

(1) f is strictly split continuous at p ∈ X;

(2) There is y ∈ Y diﬀerent from f(p) such that (a) there exists hxni → p with hf(xn)i → y, and (b) each sequence convergent to p has a subsequence along which f converges to either y or f(p). SPLIT CONTINUITY 5

1 1 Proof. (1) =⇒ (2). Since for all n ∈ N, f(Bd(p, n )) ∩ Bρ(yp, n ) 6= ∅, there is a 1 1 sequence hxni such that xn ∈ Bd(p, n ) and f(xn) ∈ Bρ(yp, n ). Thus, hxni → p and hf(xn)i → yp so that 2(a) holds with respect to y = yp. For condition 2(b) for y = yp, let hxni → p. It is sufficient to show either yp or f(p) is a cluster point of hf(xn)i. If not, ∃ε1, ε2 > 0 and ∃n1, n2 ∈ N such that ∀n ≥ n1, f(xn) ∈/ Bρ(f(p), ε1) and ∀n ≥ n2, f(xn) ∈/ Bρ(yp, ε2). Let n = max{n1, n2} and ε = min{ε1, ε2}. Then ∀n ≥ n, f(xn) ∈/ Bρ(f(p), ε) ∪ Bρ(yp, ε). This clearly violates our second condition in Definition 2.1 for split continuity. Therefore, either yp or f(p) is a cluster point of hxni and so there exists a subsequence of hf(xn)i convergent to either f(p) or yp. (1) ⇐= (2). We know from condition (a) that there exists a sequence hxni in X convergent to p such that hf(xn)i → y 6= f(p). This shows that the first condition of Definition 2.1 is satisfied. Now suppose the second condition of Definition 2.1 fails with respect to this y; there exists ε > 0, such that ∀δ > 0, f(Bd(p, δ)) 6⊆ Bρ(f(p), ε) ∪ Bρ(y, ε). Then 1 ∀n ∈ N, ∃xn such that xn ∈ Bd(p, n ), but f(xn) ∈/ Bρ(f(p), ε) ∪ Bρ(y, ε). Thus, hxni → p but hf(xn)i does not have a subsequence convergent to either y or f(p), a contradiction to condition (b) above.

We note for the record that y as described in (2) of the last result must be yp. Proposition 2.4. Let f : hX, di → hY, ρi be a function that is split continuous at p ∈ X. Then f is locally bounded at p.

Proof. Since f is split continuous at p, ∃yp ∈ Y and ∃δ > 0 such that f(Bd(p, δ)) ⊆ Bρ(f(p), 1)∪Bρ(yp, 1). This shows that f(Bd(p, δ)) lies in the union of two bounded sets, and as a ﬁnite union of bounded sets is bounded and bounded sets are stable under taking subsets, f(Bd(p, δ)) is bounded. Proposition 2.5. Let f : hX, di → R be a function that is split continuous at p ∈ X. Then f is either upper semicontinuous or lower semicontinuous at p.

Proof. Denote the usual metric on R by ρ. Suppose yp ≤ f(p). For ﬁxed ε > 0, ∃δ > 0, such that f(Bd(p, δ)) ⊆ Bρ(f(p), ε) ∪ Bρ(yp, ε). Thus, f(Bd(p, δ)) ⊆ (−∞, f(p) + ε) and so f is u.s.c. at p. If yp > f(p), a similar argument shows f is l.s.c. at p. The familiar Dirichlet function [3, p. 122] is strictly split continuous at each point and is upper semicontinuous at each rational and lower semicontinuous at each irrational. Here is a globally upper semicontinuous real-valued function f deﬁned on R that fails to be split continuous at the origin:  0 if x < 0,  f(x) = 1 if x = 0, −1 if x > 0.

3. Global Split Continuity We denote the space of globally split continuous functions between metric spaces hX, di and hY, ρi by split(X,Y ). Of course, a globally continuous function is globally ∗ ∗ split continuous. If f ∈ split(X,Y ), we deﬁne f : X → Y by f (x) := yx (x ∈ X). 6 GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON

Proposition 3.1. Let hX, di and hY, ρi be metric spaces and suppose f ∈ split(X,Y ). Then f ∗ is globally split continuous as well. Proof. Let p ∈ X be arbitrary. If f ∗ is continuous at p, there is nothing to show. Otherwise, we can find a sequence hxni convergent to p and ε > 0 such that ∗ ∗ ∀n, ρ(f (xn), f (p)) > ε. Using Proposition 2.3, choose hwni such that for each 1 ∗ 1 n ∈ N, d(xn, wn) < n and ρ(f (xn), f(wn)) < n . Since no subsequence of hf(wn)i can converge to f ∗(p), applying the same proposition, we can find a subsequence ∗ hwnk i along which f converges to f(p). It follows that hf (xnk )i also converges to f(p) which means that along some sequence convergent to p, f ∗ converges to f(p). One can show in a similar fashion that any sequence convergent to p has a subsequence along which f ∗ converges to either f ∗(p) or f(p). We can conclude that f ∗ is split continuous at p where (f ∗)∗(p) = f ∗(p) if f ∗ is continuous at p and ∗ ∗ (f ) (p) = f(p) otherwise. To reduce ugly notation in the sequel, we now adopt the notation f ∗∗ for (f ∗)∗. The following corollary makes explicit a formula established in the proof of the last proposition. We use it twice in subsequent sections. Corollary 3.2. Let hX, di and hY, ρi be metric spaces and let f ∈ split(X,Y ). Then for each p ∈ X, we have f ∗∗(p) ∈ {f(p), f ∗(p)}. We next record as a proposition a fact that was argued in the analysis of the previous section for a function that is split continuous at a point. Proposition 3.3. Let f be a globally split continuous function between hX, di and hY, ρi. Then f is continuous at p if and only if f ∗(p) = f(p). The equality f ∗(p) = f(p) is sufficient for continuity of f ∗ at p, but it is not necessary for continuity. Proposition 3.4. Let f be a globally split continuous function between hX, di and hY, ρi and suppose p ∈ X. If f ∗(p) = f(p), then f ∗ is continuous at p.

Proof. Let hxni be a sequence in X convergent to p. By Proposition 2.2, for each ∗ n ∈ N, we can ﬁnd wn ∈ X with d(wn, xn) < 1/n and ρ(f(wn), f (xn)) < 1/n. By Proposition 3.3, f is continuous at p so that hf(wn)i converges to f(p). Thus, ∗ ∗ hf (xn)i converges to f(p) = f (p) as well. To see failure of necessity, let hX, di be a metric space having a limit point p; ∗ evidently χ{p} is the zero function which is continuous at p. As a warm-up exercise, we leave the proof of the next result related to our last comment to the reader. Proposition 3.5. Let hX, di be a metric space. The following conditions are equivalent: (1) Whenever hY, ρi is a second metric space, f 7→ f ∗ is one-to-one on split(X,Y );

(2) f 7→ f ∗ is one-to-one on split(X, R); (3) Each point of X is an isolated point of the space; (4) Each globally split continuous function on X is globally continuous. SPLIT CONTINUITY 7

Proposition 3.6. Let hX, di and hY, ρi be metric spaces and suppose f ∈ split(X,Y ). Then gr(f) ∪ gr(f ∗) = cl(gr(f)).

Proof. The inclusion cl(gr(f)) ⊆ gr(f) ∪ gr(f ∗) is immediate from statement (2) of Proposition 2.2. That gr(f)) ⊆ cl(gr(f)) is trivial, while gr(f ∗) ⊆ cl(gr(f)) follows from statement (1) of the same proposition.

We close this section by considering some speciﬁc scenarios relative to global split continuity.

Example 3.7. We ﬁrst show that the characteristic function for an arbitrary subset c E of a metric space hX, di is split continuous. If p ∈ int(E) ∪ int(E ), then χE is continuous at p as it is constant in a neighborhood of p and yp = χE(p). If p ∈ bd(E), then conditions 2(a) and 2(b) of Proposition 2.3 are met with respect to yp = 1 − χE(p). We now know that χE is globally split continuous. Actually, ∗ we have argued that χE = χD where

D = int(E) ∪ bd(E)\E = int(E) ∪ E0\E.

We record several special cases for this formula.

• If E is an open subset of X, then D = cl(E); • If E is a closed subset of E, then D = int(E); • If E0 = (Ec)0 = X, that is, E is both dense and co-dense, then D = Ec (and conversely); ∗ • χE is the zero function if and only if E is a closed nowhere dense subset of X; ∗ • We have D = E, that is, χE = χE, if and only if E is a clopen subset of X. It follows from the ﬁrst two bullet points that if E is either a regular open set ∗∗ or a regular closed set, then χE = χE. ∗ ∗ While the formula χEc = 1 − χE is valid, it is not the case in general that the ∗ mapping E 7→ χE is a Boolean homomorphism from the power set of X to the functions from X to {0, 1}, where f ∨ g = max{f, g} and f ∧ g = min{f, g}. On the line, let E = (−1, 0) and let E = (0, 1); then χ∗ 6= χ∗ ∧χ∗ as the functions 1 2 E1∩E2 E1 E2 disagree at the origin. On the other hand, a characteristic function need not be the product of the star 1 1 1 operator. Let X be {0, 1, 2 , 3 , 4 ,...} as a metric subspace of the line and f be the characteristic function of the origin. If g∗ = f held, then the continuity of g at each nonzero point p forces g∗(p) = g(p) = 0, so that g∗(0) = 0 however g(0) may be deﬁned. We will show in a subsequent section that when hX, di is dense-in-itself, i.e., 0 ∗ when X = X, then for each E ⊆ X, there exists D ⊆ X with χD = χE. This is rather involved. We will also present a member f of split(R, R) that is not g∗ for any split continuous function g; of course, f cannot be a characteristic function. At the moment, we do not know how to characterize those split continuous functions on a dense-in-itself space that arise from the star operator. 8 GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON

Example 3.8. Along with the Dirichlet function, a familiar real-valued function that is traditionally thought of as pathological is the Thomae function T : [0, 1] → [0, 1] deﬁned by  1 if x = p with p ∈ and q ∈ coprime,  q q N N T (x) = 1 if x = 0, 0 if x is irrational. The assignment T (0) = 1 has a rationale; it is made to have T (x) = T (1 − x) throughout [0, 1]. This globally upper semicontinuous function is continuous at each irrational point and is discontinuous at each rational point [1, 3]. Perhaps surprisingly, T is split continuous on the entire domain. To establish this, we must show that T is strictly split continuous at each rational point p ∈ [0, 1]. Given any sequence hxni of irrationals such that hxni → p, clearly hT (xn)i → 0. By Proposition 2.3, it remains to show that each sequence convergent to p ∈ Q has a subsequence along which T converges to either 0 or T (p). Let hxni be any sequence in [0, 1] convergent to p. If xn = p for inﬁnitely many n, then there is a constant subsequence xnj = p and so hT (xnj )i → T (p). If not, let hxnk i be the subsequence formed by removing all terms of hxni equal to p. We claim that the sequence hT (xnk )i converges to 0. For each m ∈ N, let Fm be the so-called Farey sequence of order m [9, pp. 114- 116], that is, the set of all rational points in [0, 1] with reduced denominators less than or equal to m. Let n be large enough so that whenever n ≥ n, |p − xn| < min{|p − x| : x ∈ Fm \{p}}. Thus, whenever nk ≥ n, xnk 6∈ Fm. This implies

T (xnk ) < 1/m for nk ≥ n. Our reasoning of course shows that T ∗ is the zero function, and so we have produced a function with a dense set of nonzero values whose star is nevertheless the zero function. Example 3.9. Let hX, di and hY, ρi be metric spaces. We can study the following iterative scheme on split (X,Y ): for f ∈ split (X,Y ), put f1 = f and for each k ≥ 1, ∗ 1 1 1 put fk+1 = fk . We explain exactly what can happen when X = {0, 1, 2 , 3 , 4 ,...} as a metric subspace of the line. There are two mutually exclusive and exhaustive possibilities for a given f ∈ split(X,Y ):

1 (1) limn→∞ f( n ) exists;

(2) We can partition N into two inﬁnite subsets {N1, N2} and ﬁnd y0 6= f(0) 1 1 such that hf( n )in∈N1 → f(0) and hf( n )in∈N2 → y0.

1 1 1 In the ﬁrst case, we have for all k ≥ 2, fk(0) = limn→∞ f( n ) and fk( n ) = f( n ) for all k and n. In the second case, we have periodic behavior: for k even, we have 1 1 fk(0) = y0 and for k odd, we have fk(0) = f(p), while fk( n ) = f( n ) for all k and n.

4. Split Continuity And Multifunctions Whose Values Consist Of At Most Two Points The next string of results connects global split continuity of a function from hX, di to hY, ρi with the upper semicontinuity of a certain multifunction from X to Y having either one or two values at each point of the domain. SPLIT CONTINUITY 9

Proposition 4.1. Let hX, di and hY, ρi be metric spaces and let f and g be functions from X to Y . If the multifunction x 7→ {f(x), g(x)} is globally upper semicontinuous, then f and g are both globally split continuous functions. Proof. Put Γ(x) = {f(x), g(x)} (x ∈ X). By symmetry, we conﬁne our attention to f. Let p ∈ X be arbitrary. If f(p) = g(p), then easily f is continuous at p. Otherwise, f(p) 6= g(p). Given any ε > 0, since Bρ(f(p), ε) ∪ Bρ(g(p), ε) is a neighborhood of Γ(p), by upper semicontinuity there exists δ > 0 such that

d(x, p) < δ ⇒ Γ(x) ⊆ Bρ(f(p), ε) ∪ Bρ(g(p), ε), and in particular f(x) ∈ Bρ(f(p), ε) ∪ Bρ(g(p), ε), that is, the second condition of Deﬁnition 2.1 for f is satisﬁed with respect to y = g(p). 1 If for some ε ∈ (0, 2 ρ(f(p), g(p))), there exist α > 0 satisfying

f(Bd(p, α)) ⊆ Bρ(f(p), ε), then it is routine to check that f is continuous at p. Otherwise, for each ε > 0, we need to show that f(Bd(p, ε)) ∩ Bρ(g(p), ε) 6= ∅ in which case f satisfies the first condition of Definition 2.1 of split continuity at p and where y = g(p). Put 1 ε0 = min{ε, 3 ρ(f(p), g(p))}. By upper semicontinuity of Γ, we can find δ ∈ (0, ε) such that

f(Bd(p, δ)) ⊆ Bρ(f(p), ε0) ∪ Bρ(g(p), ε0).

1 Since ε0 < 2 ρ(f(p), g(p)), f(Bd(p, δ)) ⊆ Bρ(f(p), ε0) is impossible, and we get f(Bd(p, δ)) ∩ Bρ(g(p), ε0) 6= ∅. As max {δ, ε0} ≤ ε, it follows that f(Bd(p, ε)) ∩ Bρ(g(p), ε) 6= ∅.

The converse of the last proposition fails: on the line, f = χ{0} and g = −χ{0} are both globally split continuous while x 7→ {f(x), g(x)} fails to be upper semicontinuous at p = 0. As an even simpler example, if hX, di and hY, ρi are arbitrary metric spaces and if f : X → Y is globally split continuous but not continuous at p ∈ X, then x 7→ {f(x), f(x)} fails to be upper semicontinuous at p. We say that (globally split continuous) functions f and g form a split continuous pair if the associated multifunction x 7→ {f(x), g(x)} is globally upper semicontinuous. If f : X → Y and g : X → Y are both globally continuous, then f and g obviously form a split continuous pair. Our next result shows that for each globally split continuous function f, there is a second function g such that the two functions functions form a split continuous pair. Proposition 4.2. Let f be a globally split continuous function from hX, di to hY, ρi. Then f and f ∗ form a split continuous pair. Moreover, no proper submultifunction of x 7→ {f(x), f ∗(x)} whose graph contains gr(f) is globally upper semicontinuous. Proof. Fix p ∈ X and let V be a neighborhood of {f(p), f ∗(p)}. We can find ε > 0 ∗ such that Bρ(f(p), ε)∪Bρ(f (p), ε) ⊆ V . By the definition of split continuity we can ε ∗ ε find δ > 0 such that d(x, p) < δ ⇒ either ρ(f(x), f(p)) < 2 or ρ(f(x), f (p)) < 2 . But at each x ∈ Bd(p, δ) there exists a sequence hxni in the ball convergent to x such ∗ ∗ ε ∗ ∗ ε that f (x) = limn→∞f(xn), so either ρ(f (x), f(p)) ≤ 2 or ρ(f (x), f (p)) ≤ 2 . This shows that for each x in the ball, {f(x), f ∗(x)} ⊆ V as required. 10 GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON

The minimality of the multifunction follows from Proposition 3.6 and the fact that an usco map between metric spaces has closed graph (see, e.g., [2, p. 42]). In view of Propositions 4.1 and 4.2, we obtain the global split continuity of f ∗ for f ∈ split(X,Y ) which we proved directly in Proposition 3.1. Combining our last two propositions yields perhaps the most important characterization of global split continuity. Theorem 4.3. Let hX, di and hY, ρi be metric spaces and let f : X → Y . Then f is globally split continuous if and only if there exists a second function g such that x 7→ {f(x), g(x)} is a globally upper semicontinuous multifunction. Example 4.4. We now use Proposition 4.1 to produce a split continuous function f on [0, 1] for which f ∗ = T , our Thomae function. Define f as follows. At each rational point p, let f(p) = 0 and construct a sequence of irrational points hγp,ni convergent to p such that distinct rational points p1 and p2 have totally disjoint sequences hγp1,ni and hγp2,ni. To do this, assign to each rational point a unique coset of R/Q not equal to Q. The cosets are disjoint, and each is a dense set of irrationals, and thus there exists a sequence hγp,ni in [0, 1] contained in that coset convergent to p. Put f(γp,n) = T (p) ∀p ∈ Q ∩ [0, 1] and ∀n ∈ N and let f(x) = 0 otherwise. To show that f is globally split continuous, we will show that the multifunction Γ(x) = {f(x),T (x)} viewed as a multifunction from [0, 1] to R is globally upper semicontinuous. Let V ⊆ R be open. We must show that the set A := {x ∈ [0, 1] : Γ(x) ⊆ V } is open. Note that 0 ∈ Γ(x) for every x ∈ [0, 1]. This implies that if 0 6∈ V , then A = ∅. Thus we can assume that [0, b) ⊆ V for some b ∈ (0, 1]. Let E := {p ∈ Q ∩ [0, 1] : T (p) 6∈ V }. Since [0, b) ⊆ V , the set E is finite. Let c F := E ∪{γp,n : p ∈ E and n ∈ N}. Note that F = A , so to finish the proof that Γ is upper semicontinuous, we show that F is closed by proving that the set of limit points of F is exactly E. Clearly E ⊆ F 0. For any point w ∈ [0, 1] \ E there exists a neighborhood of w whose closure is disjoint from E. This follows from the fact that E is finite. Thus the neighborhood has only a finite number of points in F \ E, since each of these points is a member of a sequence converging to a point in E. But this implies that w 6∈ F 0. Finally, we must show that f ∗(x) = T (x) for each x ∈ [0, 1]. If x ∈ [0, 1] is ∗ rational, f (x) must be T (x) since f(γx,n) → T (x) 6= f(x). If x = γp,n for some p ∗ and n, then f (x) = 0, since there exists a sequence of irrationals hλni convergent to γp,n such that f(λn) = 0 for each n. Such a sequence could be found in a coset of R/Q unassigned to any rational point. Finally, if x is irrational with f(x) = 0, then Γ(x) = {0} and f is continuous at x, so that f ∗(x) = f(x) = 0 = T (x). We now present a striking result that supports the usage of the terminology “split continuous pair”. With our machinery at hand, it is simple to prove. Proposition 4.5. Suppose hX, di and hY, ρi are metric spaces and f and g form a split continuous pair of functions from X to Y . Suppose E ⊆ X and define a function h : X → Y by ( f(x) if x ∈ E, h(x) = g(x) if x ∈ Ec. SPLIT CONTINUITY 11

Then h is also globally split continuous. Proof. This is obvious if either E = X or E = ∅. Otherwise, deﬁne h0 : X → Y by ( f(x) if x ∈ Ec, h0(x) = g(x) if x ∈ E.

At each x ∈ X, {h(x), h0(x)} = {f(x), g(x)}, so by Proposition 4.1, h is globally split continuous. This result applies when f and g are globally continuous, and thus in particular when they are both constant functions. Thus, we get split continuity of characteristic functions which we also observed earlier. If we can partition the domain of a function f between metric spaces into two pieces {A, B} on which the restriction of f to each is continuous, it does not follow that f is globally split continuous. For example, such a partition exists for f : R → R deﬁned by  0 if x < 0,  f(x) = 1 if x = 0, −1 if x > 0. where A = {0} and B = (−∞, 0) ∪ (0, ∞). Of some interest over the years has been the class of Cauchy continuous functions between metric spaces hX, di and hY, di consisting of those functions mapping each Cauchy sequence in X to a Cauchy sequence in Y [5, 8, 12, 15]. This class sits between the continuous functions and the uniformly continuous functions; it is easy to see that the Cauchy continuous functions agree with the larger class of continuous functions if and only if the metric d is complete, and various characterizations of spaces for which they agree with the uniformly continuous functions can be found in [12]. As is well-known, each Cauchy continuous function deﬁned on a subset A of X into a complete metric space hY, di has a continuous extension to cl(A) [15]. We can state the following positive result as to when a function f : X → Y whose restriction to each block of a partition {A, B} is continuous must be globally split continuous provided the target space is a complete metric space. Proposition 4.6. Let {A, B} be a partition of a metric space hX, di and let hY, ρi be a complete metric space. Suppose that the function f : X → Y when separately restricted to both A and B is Cauchy continuous. Then f ∈ split(X,Y ).

Proof. Let hA be a continuous extension of f A to cl(A) and let hB be a continuous extension of f B to cl(B). We easily compute ( h (x) if x ∈ int(A) ∪ (bd(A) ∩ B), f ∗(x) = A hB(x) if x ∈ int(B) ∪ (bd(B) ∩ A), establishing global split continuity of f. 12 GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON

On the other hand, no such partition may exist for a globally split continuous function, e.g., the Thomae function on [0,1]. This follows from the next proposition (with respect to the Thomae function, take for p any rational point). Proposition 4.7. Let hX, di and hY, ρi be metric spaces and suppose f ∈ split(X,Y ). Let p be a point of strict split continuity of f, and suppose X can be partitioned into two sets {A, B} such that f A and f B are continuous. Then there is not a sequence hxni in X\{p} convergent to p such that ∀n ∈ N, f is strictly split continuous at ∗ ∗ xn and both hf(xn)i and hf (xn)i are convergent to f (p).

Proof. Suppose the domain of f can be partitioned into {A, B} so that f A and f B are continuous yet such a p ∈ X and a sequence hxni → p exist. Without loss ∗ of generality, we can suppose p ∈ A. Since hxni → p and hf(xn)i → f (p) 6= f(p), there exists n ∈ N such that ∀n ≥ n, xn ∈ B. Now since f is strictly split continuous at each xn, by either Proposition 2.2 or ∗ Proposition 2.3, ∀n ∈ N ∃hwn,ki convergent to xn with hf(wn,k)i → f (xn). For each ﬁxed n ≥ n, eventually the terms of hwn,ki must lie in A, else continuity of f B at xn would fail. Thus, for each n ≥ n, we can ﬁnd k(n) ∈ N with such that zn,k(n) ∈ A and both

1 1 d(z , x ) < and ρ(f(z ), f ∗(x )) < . n,k(n) n n n,k(n) n n ∗ From hxni → p, it follows that hzn,k(n)i → p, and since hf(xn)i → f (p), we ∗ get hf(zn,k(n))i → f (p) as well. This shows that f A is not continuous at p, a contradiction.

5. More On Split Continuity and Semicontinuity Of Real-Valued Functions Recalling the ﬁrst bullet point listed after the deﬁnition of upper envelope, if f : hX, di → R and p ∈ X are arbitrary, then

f(p) = max{limn→∞f(xn): hxni → p and hf(xn)i is convergent in [−∞, ∞]}.

Now if f ∈ split(X,Y ), then by its local boundedness, its upper and lower envelopes are real-valued. With Proposition 2.2 in mind, if f is globally split continuous, then at each x ∈ X, f(x) = max{f(x), f ∗(x)} and dually, f(x) = min{f(x), f ∗(x)}. As a result, with global split continuity, x 7→ {f(x), f ∗(x)} and x 7→ {f(x), f(x)} are the same multifunction, and so the upper and lower envelopes of f form a split continuous pair. However, neither need be the star of the other; the Dirichlet function furnishes an obvious counterexample. Our ﬁrst formal result of this section gives a basic characterization of real-valued split continuous functions, expressing the closure of the graph of such a function as the union of the graph of an upper semicontinuous function and the graph of a lower semicontinuous function. Theorem 5.1. Let f be a real-valued function on hX, di. The following are equivalent: SPLIT CONTINUITY 13

(1) f is split continuous; (2) f is locally bounded and gr(f¯) ∪ gr(f) = cl(gr(f)). Proof. (1) =⇒ (2). From Proposition 2.4, f is locally bounded. As we have just noted, ∀x ∈ X, {f(x), f ∗(x)} = {f(x), f(x)} which means that gr(f¯) ∪ gr(f) = gr(f) ∪ gr(f ∗). Condition (2) now follows from Proposition 3.6. ¯ (1) ⇐= (2). Let p ∈ X and since f is locally bounded, y1 = f(p) and y2 = f(p) are both finite. If f is continuous at p, we are done. Otherwise, (p, y1) and (p, y2) are distinct so by condition (2), (p, y1) and (p, y2) together comprise the points in the closure of the graph of f with first coordinate p. In particular, one of them is (p, f(p)). If y1 = f(p), then since f is locally bounded at p and bounded subsets of R are relatively compact, the sequential criteria for strict split continuity of Proposition 2.3 hold for y = y2, and if y2 = f(p), then they are satisfied with respect to y = y1. In statement (2) of the above theorem the condition that f be locally bounded can be replaced by the equivalent condition that the upper and lower envelopes of f be real-valued. If f : hX, di → hY, ρi is globally continuous, and g : Y → R is globally u.s.c. (resp. globally l.s.c.), then it is well-known that g ◦ f is globally u.s.c (resp. l.s.c.) as well. But in the case Y = R, if the inside function f is either globally u.s.c. or globally l.s.c. and the outside function is globally continuous, then the composition may fail to be either upper semicontinuous or lower semicontinuous at some point of X. To see this when X = R, once again define f : R → R by  0 if x < 0,  f(x) = 1 if x = 0, −1 if x > 0. and define g by ( 2x + 1 if x ≤ 0, g(x) = 1 − x if x > 0.

While f is globally u.s.c. and g is globally continuous, g ◦ f fails to be either u.s.c. or l.s.c. at the origin. Given g ∈ C(Y, R) continuous but otherwise arbitrary, split continuity of the inside function f : X → Y is exactly what is required for the composition g ◦ f to always be either l.s.c. or u.s.c. at each point of X, whatever the space Y may be. Indeed, the resolution of the question as to what conditions on the inside function guarantee semicontinuity of such a composition at each point of the domain was the provenance of our investigation of split continuity. We start with a preliminary result.

Lemma 5.2. If f : hX, di → hY, ρi is not split continuous at p ∈ X, then ∃hxni → p such that infn ρ(f(xn), f(p)) > 0 and hf(xn)i is not convergent. Proof. We construct a proof by contradiction; we show failure of split continuity at p and convergence of each sequence of the form hf(xn)i where hxni → p and infn ρ(f(xn), f(p)) > 0 are incompatible. Sequences of this form exist because f 14 GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON cannot be continuous at p; choose one and put a = limn→∞ f(xn) where of course a 6= f(p). By failure of split continuity of f at p, ∃ε1 > 0 such that ∀δ > 0, f(Bd(p, δ)) 6⊆ Bρ(f(p), ε1) ∪ Bρ(a, ε1). And so ∃hwni → p such that ∀n ∈ N, ρ(f(p), f(wn)) > ε1 and ρ(a, f(wn)) > ε1. Deﬁne hzni by z2n = xn and z2n−1 = wn for all n ∈ N. Clearly, hzni → p, and infn ρ(f(zn), f(p)) > 0, while hf(zn)i is not convergent. This is a contradiction. Theorem 5.3. Let hX, di and hY, ρi be metric spaces, let f : X → Y be a function, and let p ∈ X. The following are equivalent:

(1) f is split continuous at p; (2) For each metric space hW, µi, for each g ∈ C(Y,W ), the composition g ◦ f is split continuous at p; (3) Whenever g ∈ C(Y, R), then g ◦ f is either u.s.c. or l.s.c. at p.

Proof. (1) =⇒ (2). Let g : Y → W be globally continuous. Let yp be determined by the split continuity of f at p, and ﬁx ε > 0. Since g is globally continuous, ∃δ > 0 such that both

g(Bρ(yp, δ)) ⊆ Bµ(g(yp), ε) and g(Bρ(f(p), δ)) ⊆ Bµ(g(f(p)), ε).

By split continuity of f at p, there ∃λ > 0 such that f(Bd(p, λ)) ⊆ Bρ(yp, δ) ∪ Bρ(f(p), δ). Thus, (g ◦ f)(Bd(p, λ)) ⊆ Bµ(g(yp), ε) ∪ Bµ(g(f(p)), ε). It is routine to verify that condition (1) of Deﬁnition 2.1 is satisﬁed with respect to w = g(yp) for g ◦ f, and so the composition is split continuous at p. (2) =⇒ (3) follows from Proposition 2.5.

(3) =⇒ (1). Suppose f is not split continuous at p. From Lemma 5.2 ∃hxni → p and ε > 0 such that ∀n ∈ N, ρ(f(xn), f(p)) > ε and hf(xn)i does not converge to any point in Y . If hf(xn)i has at most one cluster point, since hf(xn)i is not convergent, there exists a subsequence hf(xnk )i of hf(xn)i which has no cluster points and so may be assumed to have distinct terms. As a result,

A := {f(p)} ∪ {f(xnk ): k ∈ N}

0 0 is a closed discrete subset of Y . Deﬁne g : A → R by g (f(xn2k )) = −1 and 0 0 0 g (f(xn2k−1 )) = 1 ∀k ∈ N, and g (f(p)) = 0. Clearly, g is continuous on A, and since A is closed, by Tietze’s extension theorem [10, p. 97], ∃g ∈ C(Y, R) whose restriction to A is g0. However, g ◦ f is neither u.s.c. nor l.s.c. at p. Now suppose hf(xn)i has two distinct cluster points a 6= b (there may be others as well); since for each index n, ρ(f(xn), f(p))) ≥ ε, we have a 6= f(p) and b 6= f(p). Again using Tietze’s theorem we can ﬁnd g ∈ C(Y, R) with g(a) = −1, g(b) = 1, and g(f(p)) = 0. Since a and b are cluster points of hf(xn)i, we conclude that g ◦ f is neither u.s.c. nor l.s.c. at p. By Proposition 2.5, a real-valued function f that is split continuous at a point p of a metric space is either u.s.c. or l.s.c. at p. This is a special case of the implication (1) ⇒ (2) of the following immediate corollary to our last result where the function g is the identity mapping.. SPLIT CONTINUITY 15

Corollary 5.4. Let f : hX, di → R be a function and let p ∈ X. The following are equivalent:

(1) f is split continuous at p; (2) Whenever g ∈ C(R, R), then g ◦ f is either u.s.c. or l.s.c. at p.

Example 5.5. To see that the composition of two split continuous functions need not be split continuous, let f : R → R be be deﬁned by f(x) = x if x 6= 0 and f(0) = 17, and let g : R → R be given by g = χQ∩[0,1] + 3χ[2,∞). While both functions are globally split continuous, g ◦ f is not split continuous at p = 0.

If f is real valued and globally split continuous, then by our composition theorem, so is αf + β whenever α and β are real numbers. However, the sum of two globally split continuous functions need not be globally split continuous. For example, on ∗ ∗ R, let f = χ[0,1]; as f = χ(0,1), clearly f + f fails to be split continuous at p = 0. The same example shows that a function into a product of metric spaces each of whose coordinate functions is globally split continuous need not be globally split continuous. If x 7→ (f(x), g(x)) were globally split continuous, then x 7→ f(x) + g(x) would be as well, applying our composition theorem with respect to (α, β) 7→ α + β as the continuous outside function. On the positive side, we observe that if f ∈ split(X, R) and g ∈ C(X, R), then their sum and pointwise product are both globally split continuous. 1 For a function f : hX, di → hY, ρi and x ∈ X, put ωn(f; x) := diamρ f(Bd(x, n )), and then deﬁne the oscillation of f at x by

ω(f; x) := infn∈N ωn(f; x) = limn→∞ ωn(f; x).

The following facts about oscillation are well-known: • f is continuous at p ∈ X if and only if ω(f; p) = 0;

• f is uniformly continuous if and only if hωn(f; ·)i converges uniformly to the zero function on X; • ω(f; ·) is an upper semicontinuous nonnegative extended real-valued function on X; • for Y = R, we have ω(f; ·) = f − f.

In the special case f = χE is a characteristic function, then ω(f; ·) = χbd(E). Notice that for the Thomae function T , we have T = ω(T ; ·). Necessary and sufficient conditions for a nonnegative extended real-valued function to be an oscillation function have been identified by Ewert and Ponomarev in 2003 [11]. The notion of the oscillation of a function on a subset of a metric space which agrees with the above definition when the subset is a singleton was introduced more recently in [6]. 16 GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON

While ω(f; ·) is real-valued and locally bounded whenever f is locally bounded, the associated oscillation function may not be globally split continuous. For example, consider the function f : R → R defined by  0 if x ∈ \ ,  R Q f(x) = 1 if x ∈ Q ∩ (∞, 0],  −2 if x ∈ Q ∩ (0, ∞). Then ω(f; ·) is not split continuous at p = 0. We aim to show that f ∈ split(X,Y ) is sufficient for ω(f; ·) to be globally split continuous as well. But first, we give an explicit formula for the oscillation function for a globally split continuous function that agrees with ω(f; p) = f(p) − f(p) when f is real valued. Lemma 5.6. Let f : hX, di → hY, ρi be a globally split continuous function. Then for all p ∈ X, ω(f; p) = ρ(f(p), f ∗(p)).

1 Proof. Fix n ∈ N. By Deﬁnition 2.1, for all ε > 0, there exists x ∈ Bd(p, n ) such that ρ(f(x), f ∗(p)) < ε, so that from the triangle inequality,

ρ(f(x), f(p)) > ρ(f(p), f ∗(p)) − ε. 1 ∗ Thus for all n ∈ N, sup{ρ(f(x1), f(x2)): x1, x2 ∈ Bd(p, n )} ≥ ρ(f(p), f (p)) and so ω(f; p) ≥ ρ(f(p), f ∗(p)). In the other direction, for all ε > 0, there exists n ∈ N 1 ε ∗ ε such that ∀x ∈ Bd(p, n ), f(x) ∈ Bρ(f(p), 2 ) ∪ Bρ(f (p), 2 ). From the triangle 1 ∗ inequaltiy, ∀x1, x2 ∈ Bd(p, n ), ρ(f(x1), f(x2)) < ρ(f(p), f (p)) + ε. It follows that ∗ ∗ ω(f; p) ≤ ρ(f(p), f (p)) and so ω(f; p) = ρ(f(p), f (p)). Theorem 5.7. Let hX, di and hY, ρi be metric spaces and let f ∈ split(X,Y ). Then its oscillation function ω(f; ·) is globally split continuous. Proof. Fix p ∈ X. If ω(f; ·) is continuous at p, there is nothing to show. Otherwise, by upper semicontinuity of the oscillation function and our last lemma, ω(f; p) = ρ(f(p), f ∗(p)) is positive. We will show that the oscillation function is strictly split continuous at p with ω∗(f; p) = 0 using our sequential criterion as described in Proposition 2.3. We must establish two things:

(a) there exists a sequence hxni convergent to p for which hω(f; xn)i → 0; (b) each sequence convergent to p has a subsequence along which the oscillation converges to either 0 or ρ(f(p), f ∗(p)).

We verify condition (b) ﬁrst. Suppose hxni is a sequence convergent to p which has no such convergent subsequence. As the oscillation function is u.s.c. at p, it is bounded in a neighborhood of p, and so by passing to a subsequence, we can ∗ assume that hω(f; xn)i converges to some α ∈ (0, ρ(f(p), f (p))). We will argue that this is incompatible with split continuity of f at p, a contradiction. Put ε := min{α, ρ(f(p), f ∗(p)) − α}. Choose n ∈ N so large that whenever ε n ≥ n, |ω(f; xn) − α| < 4 . By the deﬁnition of oscillation at xn, we can choose 1 for each n ≥ n, wn and zn such that d(xn, {wn, zn}) < n and |ρ(f(wn), f(zn)) − ε ε ω(f; xn)| < 4 so that |ρ(f(wn), f(zn)) − α| < 2 . Since hwni and hzni are both convergent to p, by taking n larger if necessary, we may assume without loss of generality that ∀n ≥ n, SPLIT CONTINUITY 17

∗ {f(wn), f(zn)} ⊆ Bρ(f(p), ε/4) ∪ Bρ(f (p), ε/4).

If both f(wN ) and f(zN ) live in the same ball, this gives α < ε, a contradic- ε ∗ ε tion, and if one lives in Bρ(f(p), 4 ) and the other in Bρ(f (p), 4 ), this gives α > ρ(f(p), f ∗(p)) − ε which is also impossible. For condition (a), simply notice that failure of continuity of oscillation at p means that there is a sequence hxni convergent to p for which no subsequence of ∗ hω(f; xn)i converges to ρ(f(p), f (p)); however, this sequence of oscillation values has a convergent subsequence which by condition (b) must have limit 0.

It is not necessary for f to be globally split continuous so that ω(f; ·) is. Consider f : R → R deﬁned by  1 if x ∈ Q \{0},  1 f(x) = 2 if x = 0,  0 if x ∈ R \ Q. Then f is not split continuous at p = 0 while ω(f; ·) is a constant function. Let f : X → Y and put A = {x ∈ X : f is continuous at x}; clearly, ω(f; ·) is continuous at each point in A, and ω(f; ·) is not continuous at each point in bd(A) ∩ Ac. For f globally split continuous, we now completely characterize the points p ∈ X at which ω(f; ·) is continuous.

Proposition 5.8. Let hX, di and hY, ρi be metric spaces, let f : X → Y be a globally split continuous function, and let p ∈ X. The following are equivalent:

∗ (1) f is continuous at p or whenever hxni → p and hf(xn)i and hf (xn)i are ∗ both convergent, then limn→∞ f(xn) 6= limn→∞ f (xn); (2) ω(f; ·) is continuous at p.

Proof. (1) =⇒ (2). If f is continuous at p, then so is ω(f; ·). Suppose f is not continuous at p. Then f(p) 6= f ∗(p) and so ω(f; p) = ρ(f(p), f ∗(p)) 6= 0. Let hxni → p; to prove (2), it suﬃces to show that this sequence has a subsequence along which ω(f; ·) converges to ω(f; p). Avoiding double subcripts, let hwni be a subsequence of hxni along which hf(wn)i ∗ and hf (wn)i are both convergent (such a subsequence exists by taking a subsequence of a subsequence because both f and f ∗ are split continuous at p). Then ∗ from (1), limn→∞ f(wn) = y1 and limn→∞ f (wn) = y2, where y1 6= y2. The ∗ deﬁnition of split continuity and Corollary 3.2 give {y1, y2} = {f(p), f (p)} and so

∗ lim ω(f; wn) = lim ρ(f(wn), f (wn)) = ρ(y1, y2) = ω(f; p) n→∞ n→∞ as required. (1) ⇐= (2). We prove the contrapositive. If condition (1) fails, then f is not continuous at p, so ω(f, p) 6= 0 and there exists hxni → p such that limn→∞ f(xn) = ∗ limn→∞ f (xn). By our last lemma, limn→∞ ω(f; xn) = 0 6= ω(f; p), making it impossible that ω(f; ·) is continuous at p. 18 GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON

6. Dense-In-Themselves Spaces And The Star Operator In Example 3.7, we presented a globally split continuous function f deﬁned on 1 1 1 ∗ {0, 1, 2 , 3 , 4 ,...} which was not g for any globally split continuous function g. Producing such a function in the context of functions whose domain is dense-in- itself takes more imagination.

Example 6.1. We use Corollary 3.2 to produce f ∈ split(R, R) that is not g∗ for any split continuous function g. We deﬁne f by

 x if x ∈ \{0},  Q f(x) = 1 if x = 0,  −x if x ∈ R \ Q. Supposing such a function g existed, then ( −x if x ∈ , g∗∗(x) = f ∗(x) = Q x if x ∈ R \ Q. By Corollary 3.2 we know

∗∗ ∗ g (x) ∈ {g(x), g (x)} = {g(x), f(x)} (x ∈ R) and since f(x) 6= g∗∗(x) for all x, we must have g(x) = g∗∗(x) for all x. Clearly, g∗(0) = 0 6= 1. We now show that on a metric space that is dense-in-itself, each characteristic function arises as the star of another characteristic function. Theorem 6.2. Let hX, di be a metric space such that X0 = X. Then for each ∗ E ⊆ X, there exists D ⊆ X for which χE = χD.

Proof. If bd(E) = ∅, then χE is continuous as both E and X\E are open and ∗ ˜ 1 ˜ so χE = χE. Otherwise, ﬁx n ∈ N, and let Zn be a 2n -net for E and Wn be a 1 ˜ ˜ 2n -net for X \ E. Let p ∈ bd(E). Then we can ﬁnd zn,p ∈ Zn and wn,p ∈ Wn 1 1 such that d(zn,p, p) < n and d(wn,p, p) < n . Let Zn = {zn,p : p ∈ bd(E)} and ˜ ˜ 1 Wn = {wn,p : p ∈ bd(E)}. Then for n + 1 we can extend Zn and Wn to 2(n+1) -nets ˜ ˜ ∞ Zn+1 and Wn+1 and construct the sets Zn+1 and Wn+1. Let Z = ∪n=1Zn and ∞ W = ∪n=1Wn. ∗ We are now ready to specify our subset D such that χD = χE:

D := W ∪ (int(E) \ Z) ∪ (bd(E) ∩ (X \ E)).

Let p ∈ bd(E) ∩ E. Then χD(p) = 0, and ∃hwmi → p such that for all m ∈ N, ∗ wm ∈ W and so χD(p) = 1. By a similar argument, for p ∈ bd(E) ∩ (X \ E), we ∗ get χD(p) = 0. 1 Now consider p ∈ int(E), and choose k ∈ N such that d(p, bd(E)) > k . Suppose 0 1 p ∈ Z . Then ∃z1, z2 ∈ Z such that d(zi, p) < 8k , for i ∈ {1, 2}. Then d(z1, z2) < 1 ˜ 2k 4k , and by the nesting of the sequence of sets hZni, {z1, z2} 6⊆ ∪j=1Zj. Without 2k 1 loss of generality, z1 ∈/ ∪j=1Zj. Still, z1 ∈ Z, so ∃u ∈ bd(E) such that d(u, z1) ≤ 2k 1 and so d(p, u) < k , a contradiction. Therefore, there exists a neighborhood Vp of SPLIT CONTINUITY 19

0 p such that Vp ⊆ int(E) and (Vp \{p}) ∩ Z = ∅ so that Vp \{p} ⊆ D. As X = X, ∗ we can ﬁnd a sequence in Vp \{p} convergent to p and so χD(p) = 1. A similar ∗ argument shows that for p ∈ int(X \ E), χD(p) = 0.

Let E be a subset of hX, di. In Example 3.7 we introduced but did not use an ∗ alternate formula for χE, namely

∗ χE = χint(E)∪E0\E We now use this formula along with the techniques of the proof of the last theorem to show that starting with a singleton subset E1 = {p} of a dense-in-itself metric space, we can construct a sequence of distinct nowhere dense subsets hEki such that for each k ≥ 2, we have χ∗ = χ . We begin with a crucial lemma Ek Ek−1 with respect to nowhere dense subsets.

Lemma 6.3. Suppose hX, di is a dense-in-itself metric space and D1 ⊆ X is nonempty, closed and nowhere dense. Then there exists D2 ⊆ X \D1, also nowhere 0 dense, such that D2 = D1 so that D1 ∪ D2 is closed.

Proof. Since D1 is nowhere dense and closed, X \ D1 is open and dense. Let Ten be 1 a 2n -net for X \ D1. Since bd(D1) = D1, for each p ∈ D1, we can choose tp,n ∈ Ten 1 such that d(p, tp,n) < n . Let Tn = {tp,n : p ∈ D1}. We can then extend Ten to a 1 ∞ 2(n+1) -net for X \ D1 and construct the set Tn+1 as we did Tn. Let D2 = ∪n=1Tn. In a similar manner to the proof of Theorem 6.2, using the fact that X0 = X, it 0 can be shown that D2 = D1. Therefore, int(cl(D2)) = int(D2 ∪ D1) = ∅, since 0 D2 ∩ D2 = ∅ and D1 is nowhere dense. Thus D2 is also nowhere dense. Theorem 6.4. Let hX, di be a metric space such that X0 = X. Then there exists a sequence of distinct, nowhere dense subsets hE i of X such that ∀k ≥ 2, χ∗ = k Ek

χEk−1 .

Proof. Let p ∈ X and put D1 = {p}, a nowhere dense closed set. By the last lemma, we can recursively construct a sequence hDki of nowhere dense subsets of k k−1 X such that for each k ∈ N, ∪i=1Di is closed, and ∀k ≥ 2,Dk ∩ (∪i=1 Di) = ∅ and 0 k−1 Dk = ∪i=1 Di.

For each k ∈ N, put Ek := ∪i∈Ik Di, where Ik = {i ≤ k : i ≡ k (mod 2)}. Now ﬁx k ∈ {2, 3, 4,...}. From our alternate formula for the star of a characteristic function, χ∗ = χ , where S = int(E ) ∪ E0 \ E . Since ∪k D is nowhere dense, Ek S k k k i=1 i

k int(Ek) ⊆ int(∪i=1Di) = ∅. As E0 = ∪k−1D , it follows that S = ∪ D = E and so χ? = χ . k i=1 i i∈Ik−1 i k−1 Ek Ek−1

We now use our backwards iteration scheme to derive a forwards iteration scheme in a dense-in-itself space that produces distinct sets, provided we start with an appropriate set that is far more complicated than a singleton.

Theorem 6.5. Let hX, di be a metric space such that X0 = X. Then there exists a sequence of distinct sets hE i such that χ = χ∗ for each k ≥ 2. k Ek Ek−1 20 GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON

Proof. Let hxji be a sequence of distinct points convergent to x0 such that xj 6= x0 for all j ∈ N. For each point xj, let Vj be an open neighborhood of xj such that (i) diam(Vj) < d(x0, xj) and (ii) cl(Vi) ∩ cl(Vj) = ∅ whenever i 6= j. Next, for each positive integer j, put Fj = cl(Vj). Since X is dense-in-itself, each Fj is dense-in-itself as well. The proof of Theorem 6.4 shows that for each j ∈ N we can construct in the metric subspace Fj a sequence of nonempty and nowhere dense subsets hEj,ni of Fj (and thus nowhere dense in X as well) satisfying the conclusions of the theorem where E = {x }. Deﬁne E := S∞ E and χ := χ∗ for j,1 j 1 n=1 n,n Ek Ek−1 each k ≥ 2. By construction, for each j ∈ N, cl(Ej,j) ∩ cl(∪n6=jEn,n) = ∅. As a result, after applying the star operator to E1, we see that F1 ∩ E2 = ∅ and Fj ∩ E2 = Ej,j−1 for each j ≥ 2. Repeating these arguments shows that in general,

( ∅ if j < k, Fj ∩ Ek = Ej,j−k+1 if j ≥ k.

This last formula shows that hEki is a sequence of distinct subsets, and arguably completes the proof. But one would like to see an actual description of each set Ek; this depends on whether k is even or odd. By requirement (i) above with respect to the diameter of each Vj, x0 is a limit ∞ point of every Ek and the only one lying outside of ∪j=1Fj. Furthermore, the fact that each Ej,n is nowhere dense in Fj implies that x0 is also a limit point c of Ek. By the alternate formula for the product of the star operator applied to a S∞ characteristic function, for each positive integer k, E2k = n=2k En,n−2k+1 ∪ {x0} S∞ and E2k−1 = n=2k−1 En,n−2k+2.

7. At what points is f ∗ continuous? In our ﬁnal section, we revisit this basic question: given a globally split continuous f between metric spaces, at exactly what points is f ∗ continuous? As we have seen, this is the case at p whenever f ∗(p) = f(p) and it also true more generally at those points p where limx→pf(x) exists. But existence of the limit of f approaching p is not necessary for continuity of f ∗ at p.

1 1 1 Example 7.1. Let E = {0, 1, 2 , 3 , 4 ,...} and let f : R → R be the characteristic function of E. Then f ∗ is the zero function, and while f ∗ is continuous at p = 0, limx→pf(x) fails to exist. Theorem 7.2. Let hX, di and hY, ρi be metric spaces and let f ∈ split(X,Y ). Suppose p ∈ X. The following conditions are equivalent:

(1) f ∗ is continuous at p;

(2) whenever hxni is a sequence in X\{p} convergent to p, there exists a se- 1 quence hwni in X such that for each n ∈ N, d(xn, wn) < n and for each ε > 0 there exists k ∈ N such that whenever n ≥ k, we have both ∗ ε ∗ ε (a) ρ(f(wn), f (xn)) < 2 and (b) ρ(f(wn), f (p)) < 2 .

Proof. (2) ⇒ (1). For (1), it suﬃces to show that whenever hxni is a sequence ∗ ∗ convergent to p such that for each n, xn 6= p, then limn→∞ f (xn) = f (p). This is immediate from condition (2). SPLIT CONTINUITY 21

(1) ⇒ (2). Let ε > 0, and by condition (1) choose δ > 0 such that d(x, p) < ∗ ∗ ε δ ⇒ ρ(f (x), f (p)) < 4 . Since hxni converges to p, we can ﬁnd k ∈ N with 1 ε k < 4 such that for all n ≥ k, d(xn, p) < δ. Using condition (1) in Deﬁnition 2.1 1 or Proposition 2.2, for each n ∈ N we can pick wn such that d(wn, xn) < n and ∗ 1 ρ(f(wn), f (xn)) < n . Then if n ≥ k, we have 1 ε ε ρ(f(w ), f ∗(x )) < < < , n n k 4 2 and so by the triangle inequality,

ε ε ε ρ(f(w ), f ∗(p)) ≤ ρ(f(w ), f ∗(x )) + ρ(f ∗(x ), f ∗(p)) < + = , n n n n 4 4 2 showing that conditions 2(a) and 2(b) both hold.

We use the last result to show the existence of limx→pf(x) is sufficient for continuity of f ∗ at p. Corollary 7.3. Let hX, di and hY, ρi be metric spaces and let f : X → Y be a 0 globally split continuous function. Suppose p ∈ X and limx→p f(x) exists. Then f ∗ is continuous at p. Proof. By Proposition 3.4, we may assume that f(p) 6= f ∗(p). There exists a sequence huni in X convergent to p such that for each n ∈ N, un 6= p and hf(un)i → ∗ ∗ ∗ f (p). From this, we conclude that limx→pf(x) = f (p). To show continuity of f at p, it suffices to show that if hxni is a sequence in X\{p} convergent to p, then we find a sequence hwni as stipulated in condition (2) of our last result. By condition (1) of Definition 2.1, we can choose a sequence hwni such that for each n ∈ N, 1 d(w , x ) < min{d(p, x ), } n n n n ∗ ε and ρ(f(wn), f (xn)) < 2 . Since hwni is a sequence in X\{p} convergent to p and ∗ ∗ ε limx→pf(x) = f (p), for all n sufficiently large, we also have ρ(f(wn), f (p)) < 2 . Apply the last theorem.

References 1. S. Abbott, Undertanding analysis, 2nd Edition, Springer, New York, 2015. 2. J.-P. Aubin and H. Frankowska, Set-valued analysis, Birkhäuser,Boston, 1990. 3. P. Bartle and D. Sherbert, Introduction to real analysis, 3rd Edition, Wiley, New York, 2000. 4. G. Beer, Topologies on closed and closed convex sets, Kluwer Acad. Publ., Dordrecht, 1993. 5. G. Beer and M.I. Garrido, On the uniform approximation of Cauchy continuous functions, Top. Appl. 208 (2016), 1-9. 6. G. Beer and S. Levi, Strong uniform continuity, J. Math. Anal. Appl. 350 (2009), 568-589. 7. C. Berge, Topological spaces, Macmillan, New York, 1963. 8. J. Borsik, Mappings preserving Cauchy sequences, Casopisˇ pˇest.Mat. 113 (1988), 280-285. 9. A. Cauchy, Exercises de mathématiques,vol. 1, de Bure Frères,Paris, 1826. 10. R. Engelking, General topology, Polish Scientific Publishers, Warsaw, 1977. 11. J. Ewert and S. Ponomarev, On the existence of ω-primitives on arbitrary metric spaces, Math. Slovaca 53 (2003), 51-57. 22 GERALD BEER, COLIN BLOOMFIELD, AND GRANT ROBINSON

12. T. Jain and S. Kundu, Atsuji completions: equivalent characterizations, Top. Appl. 151 (2007), 28-38. 13. E. Klein and A. Thompson, Theory of correspondences, Wiley, New York, 1984. 14. J. Oxtoby, Measure and category, Springer, New York, 1971. 15. R. Snipes, Functions that preserve Cauchy sequences, Nieuw Archief Voor Wiskunde 25 (1977), 409-422.

Department of Mathematics, California State University Los Angeles, 5151 State University Drive, Los Angeles, California 90032, USA E-mail address: [email protected]