ON THE THEOREMS OF Y . MIBU AND G . DEBS ON SEPARATE CONTINUITY

Using a game-theoretic characterization of Baire spaces, conditions upon the domain and the range are given to ensure a fat set C(f) of points of continuity in the sets of type X×{y}, y∈Y for certain almost separately continuous functions f:X×Y→Z. These results (especially Theorem B) generalize Mibu's. First Theorem, previous theorems of the author, answers one of his problems as well as they are closely related to some other results of Debs [1] and Mibu [2].


I. INTRODUCTION
Since the appearance of the celebrated result of Namioka, many articles have been written on the topic of separate and joint continuity, see Piotrowski  [3], for a survey Aside from an intensively studied Uniformization Problem-Namioka-type theorems, see Piotrowski  [3], questions pertaining to Existence Problem (see below) as well as its generalizations, have been asked Let X and Y be "nice" (e g Polish) topological spaces, let M be metric and let f X Y M be separately continuous, that is, continuous wit.h respect to each variable while the other is fixed Find the set C(f) of points of(joint) continuity of f.
Let us recall that given spaces X, Y and Z, and let f X x Y Z be a function.For every fixed z E X, the function f Y Z defined by f (t) f(x, ), where t Y, is called an z-secnon of f A t-section fu of f is defined similarly.
One way to ensure the existence of "many" points of continuity in X x Y can be derived from the following.Baire-Lebesgue-Kuratowski-Montgomery Theorem (see Piotrowski [3]) Let X and Y be metric and let f X Y R have all z-sections f continuous and have all t-sections fu of Baire class a Then f is of class a + 1 If a 0, that is, f,j is continuous, f is of class Now, by a theorem ofBaire, C(f) is residual So, fwe assume addtmnally that X x Y is Baire, then C(f) is a dense Ge subset ofX x Y i-I But one cannot relax the assumptions pertaining to the sections too much EXAMPLE.Let I [0, 1] and let IR be the set ofreals Put D, {(z,y) z , y , where k and p are all odd numbers between 0 and 2'} Let D t3 D, It is easy to see that C! D I Now, let us define f 19-IR by f(z, y) 1, for (z, y) E D and f(x, y) 0 if (x, y) D .all the z- sections f and all the y-sections f are of first class of Baire and C(f) 05 El However, the following three important results hold MIBU'S FIRST THEOREM (Mibu [2]) Let X be first countable, Y be Baire and such that X x Y is Baire Given a metric space M If f X x Y M is separately continuous, then C(F) is a dense G subset ofX x Y MIBU'S SECOND THEOREM [2].Let X be second countable, Y be Baire and such that X x Y is Baire Given a metric space M If f X x Y M has a) all x-sections f have their sets D(f) of points of discontinuity of the first category and, b) all y-sections f are continuous Then C(f) is a dense, Ge subset ofX x Y Following Debs [1], a function f" X M is called first class if for every > 0, for every nonempty subset A C X, there is a nonempty set U, open in A, such that diana (f(U)) <_ .
DEBS' TItEOREM 1] Let X be first countable Y be a special c-favorable space (thus Baire), X x Y be Baire Given a metric space M If f" X x Y M has: a) all z-sections f of first class-in the sense of Debs and,   b) all y-sections fu continuous Then C(f) is a dense G subset of X x Y 2. QUASI-CONTINUITY ON PRODUCT SPACES A function f X Y is called quasi-continuous at a point z X if for each open sets A C X and Hcf(X), where zA and f(z)H, we have AIntf-(H)O A function fXY is called quast-continuous, if it is quasi-continuous at each point z of X.
A function f X x Y Z (X, Y, Z arbitrary topological spaces) is said to be quasi-continuous at (p, q) X x Y with respect to the variable y, if for every neighborhood N of f(p, q) and for every neighborhood U x V of (p, q), there exists a neighborhood V' of q, with V' C V, and a nonempty open U' c U, such that for all (z, y) U' x V' we have f(z, y) N. If f is quasi-continuous with respect to the variable y at each point of its domain, it will be called quasi-continuous with respect to The definition of a function f that is quasi-continuous-with respect to z is quite similar.If f is quasi- continuous with respect to z and y, we say that f is symmetrically quast-conttnuous.
One can easily show from the definitions that if f is synunetrically quasi-continuous, then f and fu are quasi-continuous for all z X and / Y.The converse does not hold.
LEMMA (Piotrowski [4] Theorem 4 2).Let X be a Baire space, Y be first countable and Z be regular If f is a function on X x Y to Z such that all its z-sections f are continuous and all itssections fu are quasi-continuous, then f is quasi-continuous with respect to y The converse does not hold As an immediate consequence we obtain (Piotrowski [4] ..Corollary 4 3) Let X and Y be first countable, Baire spaces and Z be a regular one If f X Y Y is separately continuous, then f is symmetrically quasi-continuous If X and Y are second countable Baire spaces and Z is a regular one, and a function f:X x Y Z, then the following implications hold (which show the inclusion relations between proper classes of functions) see Diagram None of these implications can, in general be replaced by an equivalence, see Neubrunn [5] f-symmetrically quasi-continuous f-continuous

Diagram
The Banach-Mazur tame.We will use here the classical Banach-Mazur game between players A and B both playing with perfect information (see Noll  [6], Oxtoby [7]) A strategy for player A is a mapping c whose domain is the set of all decreasing sequences (G1, G9.,_1), n > 1, of nonempty open sets such that c(G1, G2,-1) is a nonempty open set contained in Gg.,_.Dually, a strategy for player B is a mapping/3 whose domain is the set of all decreasing sequences (U1, U2,), n > 0, of nonempty open sets such that/3(U1, U2,) is nonempty, open and contained in U2, Here n 0 stands for the empty sequence, for which/3(0) is nonempty and open, too.If c, /3 are strategies for A, B respectively, then the unique sequence G1,Gg.,G3,... defined by ()=GI, a(G)=G2, /3(G1, G2) G3, a(G1, G2, G3) G4, is called the game of A with a against B with/3 We will say that A with a wins against B with/3 if tq {G, n E N} holds for the game G, Gg.,... of A with a against B with/3.Conversely, we will say that B with/3 wins against A with a if A with a does not win against B with/3.
We will make use of the following theorem, essentially proved by Banach and Mazur of.Oxtoby [7], see also Noll  [6] where the game-theoretic characterization of Baire spaces was applied to obtain some graph theorems.
Let E be a topological space.The following are equivalent: (1) E is a Baire space; (2) for every strategy/3 of B there exists a strategy a of A with wins against B with/3./498 Z PI() I'ROWSKI

THE MAIN RESULT
Let us recall If A c X and b/is a collection of subsets of X, thenst.(A,ld) [.J{U U fq A :/: 0} For x E X, we write st.(z,N) nstead ofst.({x},Lt)A sequence {G,,} of open covers of X is a development of X if for each x X the set {st.(x, Gn) n N} is a base at z A developable space is a space which has a development A Moore .space is a regular developable space THEOREM A. Let X be a Baire space, Y be space and let {P, }, be a development for Z If f X Y Z is quasi-continuous with respect to y, then C(f) is a dense, G subset in X {y}, for all y E Y PROOF.Let x X, V E Y and let U V be a neighborhood of (x, y) Define a strategy for a player/3 in a corresponding Banach-Mazur game played over X For this purpose we shall order (well- ordering) the sets X, open neighborhoods of y and open nonempty subsets of X (1) B(O) has to be defined Since Z has a countable development ,, there is a local countable base at every point of Z, in particular take {G,} at f(x,) Pick G1 Now by the quasi-continuity of f with respect to y, there is a neighborhood V of y, and a nonempty open U such that f(U V c G1 Let us further assume that U and V are thefirst sets in their orderings of X and Y, respectively with the above property Now, let W be the first nonempty open set contained in U and let xl be the first element ofW Thus, W V is a neighborhood of (x,y) So, let (0) w (2) /(G, Gg)has to be defined, where G, Gg. are nonempty open and G c G Now, f is quasi- continuous with respect to y at (zg.,y), pick G3, the first element of the base at f(zt,y) with G3CG Now pick the first element U xV such that f(U xV3) C133-suchaU V exists, by the quasi-continuity with respect to y of f Now, let W 3 be the first open nonempty set contained in U 3 (a priori, it can be even the same set (!)) and let z3 U 3 be the first element of Wa.Thus, W 3 x V 3 is a neighborhood of (z3, y) So, let/(G1, G) W3.
(3) In this way we proceed to define fl by recursion, e., if/3(0) G1 and/(G, G2k) G2k+l, for all k < n then the former steps are available and we can define G2k+a in analogy with (2).
(4) Suppose now that has been defined.Since X is Baire, there is a strategy a for A such that A with a wins against B with (see the definition of the game).
Let G1, G2... be the game A with a against B with Notice that.
N{ o N} N{ o N}. ( But observe that a is winning, hence this intersection is nonempty; i.e, x* 5 I'I{W,, :n N}, so (x',y) (U V) t2 (X {y}).This in turn shows the density of C(f) in X {y} The G6 part follows easily from the construction I-I A space will be called quast-regular if for every nonempty open set U, there is a nonempty open set V such that CI V c U Obviously, every regular space is quasi-regular.
Let .A be an open covering of a space X Then a subset S of X is said to be A-small if S is contained in a member of .,4A space X is said to be strongly countably complete if there is a sequence of open coverings of X such that a sequence {F,} of closed subsets of X has a nonempty intersection provided that F, F,. for all and each F, is .A,-small The class of strongly countably complete spaces includes locally countable compact spaces and complete metric spaces  In view of the following (Piotrowski [8], Theorem 4 6 see also Lemma 3 of Piotrowsk [9]) Every quasi-regular, strongJy countably complete space X is a Baire space Theorem A is a strong generalization of the following (Piotrowski [8], Theorem 4 5) Let X be a space, Y be quasi-regular, strongly countably complete and Z be metric If f X Y Z is quasi-continuous with respect to z, then for all z E X the set of points ofjoint continuity of f is a dense G of {z} Y Further, observe that our Theorem A answers, in positive, the following (Piotrowski [8], Problem 4 11) Does Theorem 4 5 (of [8]) hold if Y is only assumed to be a quasi- regular Baire space9 The following Theorem B is the main result of this paper and its proof easily follows from the lemma and Theorem A THEOREM B. Let X be first countable, Y be Baire and Z be Moore If f X x Y Z has all its z-sections fx quasi-continuous and all its v-sections fy continuous, then for all z E X, the set of points of continuity of f is a dense G subset of z Y The above result strongly generalizes (see the assumptions upon Y and Z) the following known theorem (Piotrowski [8], Theorem 4 8 see also Theorem 5 of Piotrowski [9]) Let X be first countable, Y be strongly countably complete, quasi-regular and Z be a metric space If f X Y Z is a function such that all its z-sections fx are quasi-continuous and all its v-sections fy are continuous, then for all z X, the set of points ofjoint continuity of f is a dense G, subset of z } Y Our Theorem B generalizes in many ways Mibu's First Theorem-see Introduction It is also closely related to Mibu's Second Theorem and Debs' Theorem ibidem Observe though, that quasi-continuity of a function does not imply nor is implied, by the condition of being of first class-in the sense of Debs Really, let f [0, 1] ]R be given by f(z) 0, if z #-1/2.Then such a function f is of first class, in the sense of Debs and, clearly, it is not quasi-continuous There are quasi-continuous functions f:li-R which are of arbitrary class of Baire or not Lebesgue measurablesee Neubrunn [5] for more details REMARK 1.The studies of the continuity points of functions whose ranges are not necessarily metric have been done already in the 1960's, see Klee and Schwarz [10] or later in the 1980's, see Dubins 11 ], we omit here an extensive literature of this approach, when the range is a uniform space REMARK 2. Recently, the author has obtained some results of this paper using though entirely different techniques, see Piotrowski 12]