ANOTHER NOTE ON KEMPISTY ’ S GENERALIZED CONTINUITY

.ABSTRACT. Under a fairly mild completeness condition on spaces Y and Z we show that every x-continuous function f: X Y Z M has a "substantial" set C(f) of points of continuity. Some odds and ends concerning a related earlier result shown by the authors are presented. Further, a generalization of S. Kempisty’s ideas of generalized continuity on products of finitely many spaces is offered. As a corollary from the above results, a partial answer to M. Talagrand’s problem is

x-COnO if for every (p,q,r) g X Y Z, for every neighborhood U V W of (p,q,r) and for every neighborhood N of f(p,q,r) there exists a neighborhood U' of p with U'c U and nonempty open sets V' and W' with V'c V and W'c W such that for all (x,y,z) E U' V' W' it follows that f(x,y,z) E N.
We shall first show that under certain general assumptions concerning the spaces, x-continuous functions have "large" sets of points of joint continuity.In order to do this we first list some necessary definitions.
Let A be an open covering of a space X.Then a subset S of X is said to be A-6m2 if S is contained in a member of A. A space X is called ongy e0b complete if there exists a sequence {Ai: i=1,2   of open coverings of X such that empty intersection.
The class of strongly countably complete spaces include countably compact and complete metric spaces.This fact follows easily from a theorem due to A.
Arhandel'skii [3] and Z. Frolk [4] which states that in the class of completely regular spaces, ech-complete and strongly countably complete spaces coincide (Engelking [5]), see also Frolk [4] where some other properties of these spaces such as their invariance under taking closed open subspaces or products are discussed.
A space X is called quasi-regular, (Oxtoby [6]) if for every nonempty open set u, there is a nonempty open set V such that clV c u. Obviously, every regular space is quasi-regular.
Let us recall that a function f: X y Z is said to be asi-coOS with repecZ to x, (Kempisty [I] p.188,) if for every (p,q) X y, fore very neighbor- hood N of f(p,q) and every neighborhood U V of (p,q) there exists a neighborhood U' of p with U' c U and a nonempty open set V' c V such that for all (x,y) U' V' we have f(x,y) N. Quasi-continuity with respect to y can be defined similarly.
LEMMA I. (Lee and Piotrowski [2] Lemma 3 p. 383).Let X, Y, Z and T be spaces and let F: X Y Z T be a function.Then f is x-continuous if and only if g: X S T is quasi-continuous with respect to x, where S Y Z and g(x,(y,z)) f(x,y,z).
THEOREM 2. Let X be a space, Y and Z be spaces such that Y Z is quasi-regular, strongly countably complete and let M be metric.If f: X Y Z M is x-continuous then for every x E X, the set C(f) of continuity points of f is dense G subset in {x} Y Z.
PROOF.In view of Lemma it is sufficient to prove the following: CLAIM.Let X be a space Y be a quasi-regular strongly countably complete and Z be metric.If f: X y Z is quasi-continuous with respect to x, then for all x g X the set of points of joint continuity of f is a dense G subset of {x} Y.
PROOF.First we will prove that the set of points of joint continuity of f is dense in {x} y.Let x E X, y Y and U V be any neighborhood U of x, contained in U, and a nonempty open set V c V such that for all (x',y') and (x",y") in U V I, we have p(f(x',y'), f(x",y")) < I. Without loss of generality we may assume that V is contained in an element A of the covering A of Y. Let W be a nonempty open set such that cl W = V I.So cl W is Al-small.Then U W is a neighborhood of (x,Yl), where u W I, and since f is quasi-continuous with respect to x at (x,Yl), there is a neighborhood U 2 of x, contained in U and a nonempty open set V 2 W I, such that for all (x',y') and (x",y") in U V we have p(f(x',y'), f(x",y")) < 1/2.Similarly, we may assume that V = is contained in an element A 2 of the covering A 2. Let W = be a nonempty open set such that cl W 2 V2.We see, that cl W 2 is A2-small.The proof that this set is G 6 subset of {x} y easily follows, when we recall that the function f takes values in the metric space Z.This completes the proof of Claim.
Thus, Theorem 2 is shown.
The forthcoming, Proposition 3 is contained in Lemma 5.1 of [6], since any quasi-regular strongly countably complete space is pseudo-complete; take B(n) the class of all nonempty open sets that are A -small.Then {B(n)} is a n sequence of (pseudo-) bases that shows X to be pseudo-complete.)We would like to thank the referee who make the above observation.PROPOSITION 3. (Oxtoby [6], Lemma 5.1) Every quasi-regular strongly countably complete space X is a Baire space.
REMARK 4. Observe that neither base countability nor metrizability assumptions are made on the considered spaces X, Y, Z in Theorem while in Theorem 2 of [2] the same conclusion concerning the set of points of continuity is obtained under an g assumption that X is first countable, Y is Baire, Z is second countable in a neighborhood of any of its points and such that Y Z is Baire.

CONDITIONS IMPLYING x-CONTINUITY
COUNTER-EXAMPLES.
Given spaces X and Y; a function f: X Y is said to be quasi-continuous (Martin [8], compare Kempisty [l]) if for every x e X and for every neighborhood U of x and for every neighborhood V of f(x) have: U N Int f (V) # .
The main result of Lee and Piotrowski [2] is the following: THEOREM A. (Lee and Piotrowski [2], Theorem I, p. 383).Let X be first count- able, Y be Baire, Z be second countable such that Y Z is Baire and let T be regular.If f: X y Z T is: (I) continuous at X {y} {z}, y Y, z Z, and (2) quasi-continuous at points of {x} y {z} for all x e X and z Z, and (3) quasi-continuous at points of {x} {y} Z for all x E X and y Y then f is x-continuous.
The first natural question which comes up is to check whether the converse of Theorem A is true.Apparently, the following Example 5 settles this question in the negative.

0, otherwise
The function f is x-continuous, however, fixing y 0 z we obtain that f(x,0,0) is not continuous.Now we shall investigate the necessity of the assumptions in Theorem A, in particular: (*) c0ti/ty of f at points of X {y} {z} (**) qu%-eouy of f at points of {x} Y {z}, and (***) qu%-eoubt of f at points of {x} {y} Z.In what follows (Examples 6 and 7) such constructions will be provided.EXAMPLE 6.The assumption (*) is essential.In fact, let us consider a func- tion f: [-1,1] 3   3 given as follows (x,y,z+l), if (x,y,z) e [0,I] [0,I] [0,I] f(x,y,z) (x,y,z-l), if (x,y,z) e [-I,0] [-I,0] [-I,0] (x,y,z), otherwise A standard verification that f has the required property (namely f is not x-continuous at (0,0,0)) is left to the reader.Using somewhat more complex, but still elementary techniques we shall show that also (**) (as well as (***)) is essential.In fact, we have EXAMPLE 7. Consider the function g:[-l,l] 3   3 given as follows: (x,y,z + I) if (x,y,z) g [-I,i]  [-,I] Again, we leave to the interested reader a standard verification that f is not x-continuous at (0,0,0).

ONE-PROMISING HYPOTHESIS.
Observe that the definition of x-continuity at (p,q,r) requires the existence of a "small" neighborhood U' of p and "small" nonempty open sets V' and W' such that q and r "clusters" to V' and W' respectively and such that the set f(U' V' W')   is contained in a "small", previously chosen, open set N. This observation prompts us to label this kind of product almost continuity as 1-5-e0btinu/g since we require the existence of only one "small" neighborhood U' (around p) of the three neighborhoods U, V, W.
The term "1-3-continuity" has been used already, in a different sense in Breckenridge and Nishiura [9].So, now let us consider "2-3-continuity".More precisely, given spaces X, Y, Z and T, we say that f: X Y Z T is f-3-COR0u or more specifically xg-colcZi.naOU.6,if for every (p,q,r) e X y Z, for every neighborhood U V W of (p,q,r) and for every neighborhood N of f(p,q,r) there is a neighborhood U' of p, with U' U, there is a neighborhood V of q, with V Ic V and a nonempty open set W I, with W Ic W such that for all (x,y z) e U x V W we have f(x,y,z) e N. Now, 3-3-continuity can be defined easily; the set W in definition of 2-3-continuity is assumed to be a neighborhood of r not just only a nonempty open subset of W.
It now follows from a result of T. Neubrunn [I0] that if X, Y, Z are "nice" (e.g.Baire, second countable), T-regular then if f: X y Z T is separately quasi-continuous then it is (jointly) quasi-continuous.
We can present this fact in the following symbolic equality: "0 + 0 + 0 0", where the numbers (0 or I) on the left side of the equality stand for quasi-continuity (0) or continuity (I) of the corresponding sections and the numbers on the right (i 0, I, 2 or 3) denote the corresponding i-3-contlnuity of f as a function of three variables.
Theorem A implies that if X, Y, Z and T are as above and if f: X y Z T is continuous in x and is quasi-continuous in y and is quasi-continuous in z, then f is l-3-continuous.Consequently, we get: "I + 0 + 0 i".
In view of the above considerations it is now natural to state the following: HYPOTHESIS.Let X, Y and Z be Baire, second countable spaces and let T be regular.If f: X y Z T is: I) continuous in x, and 2) continuous in y, and 3) quasi-continuous in z, Then f is 2-3-continuous; In other words: "I + + 0 2" We shall resolve this Hypothesis in the nggavg in the forthcoming Example 8.
Take f: 3 to be f(xl,x2,x3) h(x3), where h is any function which is continuous except for 0.
Using the above pattern the reader will easily construct 0-3-continuous function (E quasi-continuous) which is not l-3-continuous.
Apparently, the above constructions can be illustrated with the following very specific formula-ready example.Then f is i-3-continuous which is not (i + l)-3-continuous, i 1,2.
Having defined I-3 and 2-3-continuity for f: X X2 X3 T, we shall now extend these ideas to a general case.
Namely, let n be an arbitrary natural number.We say that f function n n f: i X i T is A-n-continuo if for every (PI' P2 Pn i X i and for every neighborhood U U 2 Un of (pl, P2' pn and for every neighborhood N of f(Pl' P2' pn there are neighborhoods U (i s k) of the first k out of n ,s with U' c U and there are (n-k) nonemptYkopen sets V' points PI' P2''''' Pn An interested reader will easily observe that the formula k i where f: n describes a k-n-contlnuous function f given by n f(x Xn) gk(Xl Xk), k I, 2, 3 n-l.
One can also give analogues of Example 8 and 9 for k-n-continulty.
Studies of C(f) in hyperspaces for separately continuous functions and related ones were done also in Bgel [II] and Hahn [12]. 5.A PARTIAL SOLUTION TO A PROBLEM OF M. TALAGRAND.M. Talagrand ([13] Problem 3 p. 160)asked whether if X is Baire, Y is compact and f: X y is any separately continuous function, is there the set C(f) of points of continuity of f nonempty.
We shall answer this question in the positive if a compact space Y is additionally In fact, we have shown the following result: LEMMA II. (Lee and Piotrowskl [2], Lemma 2 p. 381).Let X be Baire, Y be first countable and Z be regular.If f: X Y Z is a function such that all its x-sections fx are continuous with the exception of a first category set, and all its y-sections fy are quasi-continuous, then f is quasi-continuous with respect to y.
It follows from the definition that REMARK 12. Every quasi-contlnuous function with respect to y is quasl-continuous.LEMMA 13. (Marcus [14]).Let X be a Baire, M be metric.If f: X M is quasi- continuous, then C(f), the set of point of continuity of f is dense G 6 subset of X. PROPOSITION 14.Let X be Baire, Y be compact first countable and let f: X Y--]R be any separately cont[nuous function.Then C(f) # O. PROOF.By Lemma II and Remark 12 such f is quasi-continuous.Now, since the Cartesian product of a compact space and a Baire space is Baire, we are done by Lemma 13.
x,y n) Yn such that for all (x',y') and (x" y") in U n V n we have (f(x',y'), f(x",y")) < n and that V n is contained in an element A of the covering of Y.Moreover, there n n is a nonempty open sets W n such that V n+l cl W n Vn.Thus each cl W n is A -small n obviously cl W n m cl Wn+l.Since Y is strongly countably complete cl ,y*) g (U V) ({x} y) and (x,y*) is a point of joint continuity of f.This shows the density of the set of points of joint continuity of f in the set {x} xy.

2 Xn
such that for all (x I, x