View source for Separate and joint continuity

It follows from a property of the [[product topology]] that every [[continuous function]] $f:X\times Y\to Z$ between [[topological space]]s is separately continuous, i.e., $f$ is continuous with respect to each variable while the other variable is fixed [[#References|[a3]]]. It was observed by E. Heine [[#References|[a14]]], p. 15, that, in general, the converse does not hold (see also [[#References|[a11]]] for an account of early discoveries in this field).

Given  "nice"  topological spaces $X$ and $Y$, let $M$ be a [[metric space]] and let $f:X\times Y\to M$ be separately continuous. Questions on separate and joint continuity are, among others, problems of the type:

the 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 (cf. [[Set of type F sigma(G delta)|Set of type $F_\sigma$ ($G_\delta$)]]) of $X\times Y$. For example, every real-valued separately continuous function $f:\textbf{R}^2\to\textbf{R}$ is of the first Baire class (cf. also [[Baire classes]]), hence $C(f)$ is a dense $G_\delta$-subset.

There is also interest in a  "fibre"  version; it is the same as above, except now one looks for $C(f)$ in $\{x\}\times Y$, for any fixed $x\in X$.

The characterization problem. Characterize $C(f)$ as a subset of $X\times Y$

If $X=Y=M=\textbf{R}$, then the set $C(f)$ is the complement of an $F_\sigma$-set contained in the product of two sets of the first Baire category [[#References|[a8]]].

The uniformization problem. Find out whether there is a dense $G_\delta$-subset $A$ of $X$ such that the set $A\times Y$ is contained in $C(f)$.

If $X=Y=M=\textbf{R}$, such a result was known already to R. Baire [[#References|[a1]]]. If $X$ is a complete metric space, $Y$ is a compact metric space and $M=\textbf{R}$, the uniformization problem was positively solved by H. Hahn [[#References|[a7]]]. I. Namioka [[#References|[a9]]] extended Hahn's result to $X$ being a regular, [[strongly countably complete topological space]], $Y$ being a locally compact $\sigma$-compact space and $Z$ being a pseudo-metric space.

Following [[#References|[a4]]], one says that $X$ is a ''[[Namioka space]]'' if for any compact space $Y$, any metric space $M$ and any separately continuous function $f$, the uniformization problem can be positively solved. M. Talagrand [[#References|[a13]]] constructed an $\alpha$-favourable space (hence, a Baire space) that is not Namioka. J. Saint Ramond [[#References|[a12]]] proved that [[Separable space|separable]] [[Baire space]]s are Namioka, Namioka [[Tikhonov space]]s are Baire, while in the class of metric spaces, the set of Namioka spaces and the set of Baire spaces coincide.

A space $Y$ is a co-Namioka space if for any Baire space $X$, any metric space $M$ and for any separably continuous function $f$, the uniformization problem can be positively solved. For example, Corson-compact spaces are co–Namioka, whereas the Čech–Stone compactification $\beta N$ is not. Many results in this direction have been obtained by R. Haydon, R.W. Hansell, J.E. Jain, J.P. Troallic, Namioka, and R. Pol (see [[#References|[a10]]] for a comprehensive exposition of this topic, organizing research in this field until the middle of the 1980s).

==Applications.==
R. Ellis [[#References|[a5]]], [[#References|[a6]]] showed that every locally compact semi-topological group (i.e., a 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 [[#References|[a2]]] extended Ellis' theorem to all Čech-analytic Baire semi-topological groups (hence, to all Čech-complete semi-topological groups).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Baire,   "Sur les fonctions des variables réelles"  ''Ann. Mat. Pura Appl.'' , '''3'''  (1899)  pp. 1–122</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Bouziad,   "Every Čech-analytic Baire semitopological group is a topological group"  ''Proc. Amer. Math. Soc.'' , '''124'''  (1996)  pp. 953–959</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.L. Cauchy,   "Cours d'Analyse de l'Ecole Polytechnique"  (1821)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.P.R. Christensen,   "Joint continuity of separately continuous functions"  ''Proc. Amer. Math. Soc.'' , '''82'''  (1981)  pp. 455–462</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Ellis,   "A note on the continuity of the inverse"  ''Proc. Amer. Math. Soc.'' , '''8'''  (1957)  pp. 372–373</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R. Ellis,   "Locally compact transformation groups"  ''Duke Math. J.'' , '''24'''  (1957)  pp. 119–125</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Hahn,   "Reelle Funktionen" , Akad. Verlag  (1932)  pp. 325–338</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  R. Kershner,   "The continuity of functions of many variables"  ''Trans. Amer. Math. Soc.'' , '''53'''  (1943)  pp. 83–100</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  I. Namioka,   "Separate and joint continuity"  ''Pacific J. Math.'' , '''51'''  (1974)  pp. 515–531</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  Z. Piotrowski,   "Separate and joint continuity"  ''Real Anal. Exch.'' , '''11'''  (1985-86)  pp. 293–322</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  Z. Piotrowski,   "The genesis of separate versus joint continuity"  ''Tatra Mtn. Math. Publ.'' , '''8'''  (1996)  pp. 113–126</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J. Saint Raymond,   "Jeux topologiques et espaces de Namioka"  ''Proc. Amer. Math. Soc.'' , '''87'''  (1983)  pp. 499–504</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  M. Talagrand,   "Espaces de Baire et espaces de Namioka"  ''Math. Ann.'' , '''270'''  (1985)  pp. 159–164</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  J. Thomae,   "Abriss einer Theorie der complexen Funktionen" , Louis Nebert Verlag  (1873)  (Edition: Second)</TD></TR></table>