© 1996, Topology Atlas

It is well known (see e.g. [En] p.59) that every continuous function f from a space X into a Hausdorff space Y is uniquely determined by its values on any dense subset of X; in other words, if two continuous functions agree on a dense subset of X, then they agree throughout X.

Borel functions or Baire functions of any class fail to have the above property. (Consider step functions.) Also, connected functions are not determined by their values on dense subsets. (Consider two ``topological sine curves'' having different values at the origin.)

Nevertheless, W. Sierpi\'nski [Si] showed that any real-valued, separately continuous function on Rⁿ is uniquely determined by its values on any dense subset the domain space. Sierpi\'nski's result has been proven again in [Ma] and [To] and generalized by [GN] and [Co].

The following seven questions come naturally while considering possible extensions of Sierpi\'nski's result.

1. Pertaining the size of the domain, does the therorem hold for the infinite product [0, 1]^aleph₀?
2. Relating to the type of the spaces considered in the product, there are the following four cases:
   1. Is the assumption that at least one of the factors of the domain space is Baire necessary?
   2. Can the domain space be the product of two Baire spaces, at least one of which is second countable?
   3. Can the domain space be the product of a Baire space and a compact Hausdorff space?
   4. Can the domain space be the product of two compact Hausdorff spaces?
3. Pertaining to the range, is the assumption that the range space Z is the set R of the real numbers necessary?
4. Relating to the type of all x-sections f_x and all y-sections f_y: Assume that all x-sections and all y-sections are determined by their values on dense subspaces. Is the function f:R² --> R uniquely determined by its values on any dense subset?

Here are some answers to the above questions.

Ad (1). Sierpi\'nski's theorem fails for the infinite product [0,1]^aleph₀. (See [Co], Remark 2.3, p. 133.)

Ad (2.1). Yes, see [GN], p. 998, where it is stated that the theorem fails for the domain space Q x Q, where Q is the space of rational numbers.

Ad (2.2). Yes, see [Co], Corollary 2.1, p. 132, or [GN], Theorem 1, p. 997. R. A. McCoy [MC] announced another related result: Let X and Y be metric spaces, at least one of which is a Baire space, and let Z be a Hausdorff space. If f:X x Y --> Z is a separately continuous function, then f is uniquely determined by its values on a dense set.

Ad (2.3). We do not know the answer to this question. However, the problem whether every separately continuous real-valued function defined on a product of a Baire space X and a compact Hausdorff space is determined by its values on a dense subspace is naturally linked to the still unsolved problem of M. Talagrand [Ta1].

Talagrand's problem: Let X be Baire, Y be compact Hausdorff, and let f:X x Y --> R be separately continuous. Is the set C(f) of points of continuity nonempty?

Talagrand's example of an alpha-favorable space that is not Namioka provides an example of a real-valued separately continuous function defined on a product of a Baire space and a compact Hausdorff space still having ``many'' points of continuity ([Ta2], see also [P1] and [PW1]).

Let us recall that a function f:X --> Y is said to be feebly continuous if for every open nonempty subset V of Y the following holds:

A positive answer to the above question would solve both Talagrand's problem and our question (2c). Obviously, we could then use the Structural Lemma of [PW2]: in the case of Talagrand's problem, the domain X x Y is a Baire space; f, being feebly continuous, has a nonempty set C(f) of points of continuity. See also the overview [Bo].

Ad (2.4). The answer is yes. (See [Tr].)

Ad (3). No, see either [Co], [GN], or [MC], where R is replaced by spaces Z that are completely regular, regular Hausdorff, or Hausdorff, respectively.

Ad (4). No, see [PW2].

References

[Ba] Baire, R., Sur les fonctions des variables r\'eelles, Ann. Mat. Pura Appl. 3 (1899), 1--122

[Bo] Bouziad, A., Continuite jointe et continuite separee, Research Topics in Topology, Topology Atlas

[Co] Comfort, W. W., Functions linearly continuous on a product of Baire spaces, Boll. Un. Mat. Ital. 20 (1965), 128--134

[Du] Dugundji, J., Topology, Wm. C. Brown Publishers, Dubuque, 1989

[En] Engelking, R., General Topology, PWN, Warszawa, 1977

[Fr] Frol\'\i k, Z., Remarks concerning the invariance of Baire spaces under mappings, Czech. Math J. 11 (1961), 381--385

[GN] Goffman, C., Neugebauer, C. J., Linearly continuous functions, Proc. Amer. Math. Soc. 12 (1961), 997--998

[Ma] Marcus, S., On functions continuous in each variable, Doklady Akad. Nauk SSSR, vol. 112 (1957), 812--814

[MC] McCoy, R. A., Separately continuous functions, Amer. Math. Monthly 85 (1978), 199

[Nb] Neubrunn, T., Generalized continuity and separate continuity, Math. Slovaka 27 (1977), 307--314

[Ng] Neugebauer, C. J., A class of functions determined by dense sets, Archiv der Mathematik 12 (1961), 206--209

[P1] Piotrowski, Z., Separate and joint continuity, Real Analysis Exchange 11 (1985--86), 293--322

[P2] --------, Separate continuity, Namioka and Sierpi\'nski spaces (in preparation)

[PW1]} --------, Wingler, E., A note on continuity points of functions, Real Analysis Exchange 16 (1990--91), 408--414

[PW2] --------, --------, On Sierpi\'nski's theorem on the determination of separately continuous functions (submitted)

[Si] Sierpi\'nski, W., Sur une propertie de fonctions de deux variables r\'eeles, continues par rapport \`a chacune de variables, Publ. Math. Univ. Belgrade, vol.1 (1932) 125--128

[T1] Talagrand, M., Espaces de Baire et espaces de Namioka, Math. Ann. 270 (1985) 159--164

[T2] Talagrand, M., (private communication)

[To] Tolstov, G., On partial derivatives, Amer. Math. Soc. Transl. No. 69, Izv. Akad. Nauk SSSR Mat., vol. 13 (1949), 425--449

[Tr] Troallic, J.-P., Quasi-continuite, continuit\'e separ\'ee et topologie extremale, Proc. \hfill\break Amer. Math. Soc. 110 (1990), 819--827