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SPLIT CONTINUITY 1. Introduction 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! ff(x); g(x)g 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 2 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! ff(x)g. 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; fx 2 X : Γ(x) ⊆ V g 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! ff(x); g(x)g 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! ff(x); g(x)g, 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! ff(x); f ∗(x)g 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 char- acteristic 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 semicon- tinuous 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 semiconti- nuity 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 2 X, we put d(x; A) := inffd(x; a): a 2 Ag. 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 "-discrete provided whenever e1 and e2 are distinct members of E, we have d(e1; e2) ≥ ". Using Zorn's lemma, it is easy to show that each "-discrete subset E of A is contained in a maximal "-discrete subset C of A, in which case 8a 2 A; 9c 2 C with d(c; a) < ": We call a maximal "-discrete subset of A an "-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 ! (−∞; 1] is upper semicontinuous (u.s.c.) at p 2 X if whenever hxni converges to p, we have f(p) ≥ lim supn!1 f(xn). For a real-valued function f, (i) upper semicontinuity at p means that for each " > 0 there exists δ > 0 such that d(x; p) < δ implies f(x) < f(p) + ", 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) := inf">0 supff(w): d(w; x) < "g (x 2 X): The following facts about the upper envelope of a real-valued function f are all well-known: • for each p 2 X there exists a sequence hxni convergent to p for which f(p) = limn!1 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 ! [−∞; 1) is lower semicontinuous (l.s.c.) at p 2 X if whenever hxni converges to p, we have f(p) ≤ lim infn!1 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) := sup">0 infff(w): d(w; x) < "g (x 2 X): It is routine to show that at each p 2 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) := f(x; y): x 2 X; y 2 Γ(x)g: 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).
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