Given "nice" topological spaces X and Y, let M be metric and let f : X x Y -> M be separately continuous.



1. Existence Problem. Find the set C(f) of points of continuity of f. If X and Y are "nice", then C(f) is a dense G-delta subset of X x Y. For example, every real-valued separately continuous function f : R^2 -> R is of the first class of Baire, hence C(f) is a dense G-delta subset.
   There is also interest in a "Fiber" version: It is the same as above, except now we look for C(f) in {x} x Y, for any fixed x in X.
   
   
2. Characterization Problem. Characterize C(f) as a subset of X x Y. If X = Y = M = R, then the set C(f) is the complement of an F-sigma set contained in the product of two sets of first category (R. Kershner, (1943)).
   
   
3. Uniformization Problem (Namioka-type theorems). Find whether:

   (*) there is a dense G-delta subset A of X such that the set A x Y is contained in C(f).

Following J.P.R. Christensen (1981) we say that a space X is Namioka if for any compact space Y and any metric space M and any separately continuous function f, (*) holds. M. Talagrand (1985) constructed an alpha-favorable space (hence Baire) which is not Namioka. J. Saint Raymond (1983) proved that: Separable Baire spaces are Namioka, Tychonoff Namioka spaces are Baire, and in the class of metric spaces Namioka and Baire spaces coincide.

A space Y is co-Namioka if for any Baire space X and any metric space M and for any separately continuous function f, (*) hols. For example, Corson-compact spaces are co-Namioka, whereas beta-N (the Stone-Cech compactification of N) is not. Many results in this direction have been obtained by R. Haydon, R.W. Hansell, J.E. Jain, J.P. Troallic, I. Namioka and R. Pol. My (1986) expository article brings a compehensive presentation of this topic organizing research in this field until mid-80's.

Applications. R. Ellis (1957, 1957) showed that every locally compact semitopological group (e.g. group endowed with a topology for which the product is separately continuous) is a topological group. Using methods of separate and joint continuity A. Bouziad (1996) extended Ellis theorem to all Cech-analytic Baire semitopological groups (hence all Cech-complete semitopological groups).

In my talk I will provide the necessary background and formulate some still open questions in this area, e.g., pertaining to the types of sigma-ideals for which "soft" (i.e., Y is assumed to be second countable) Namioka-type theorem holds, continuity of homomorphisms (generalizations of Dudley's theorem), strengthening of Mibu-type theorems, etc.