Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)

CINCURA, J., SALÁT, T. and VISNYAI, T., On separately continuous functions R

Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) H ON SEPARATELY CONTINUOUS FUNCTIONS f: t

2 R J. Cincura, T. Salát, T. Visnyai (Bratislava, Slovakia) Abstract. In this paper the notions of separately continuous and strongly separately continuous functions R are introduced and properties of such functions are investigated. The obtained results are compared with the corresponding known results for functions defined on R

M (m>'2). It is shown that there are several interesting and essential differences between properties of (strongly) separately continuous functions defined on I

2 and properties of (strongly) separately continuous functions defined on R

M

. Introduction Separately continuous functions /: R

m R were investigated in several papers (see e.g. [2], [4], [8], [11]). Recall that a function /: R

m -» R is said to be separately continuous at a point xq = (x®,

..

., x®

}

) £ R'" provided that for each k = 1,2, ...,m the function <£>fc:R —» R defined by fk(t) = /(xi,

..., x®_ v /, x®

+,,..., x

(

1

>

l

l

) is continuous at x®. It is well known that a function can be separately continuous at x° without being continuous at x". The standard example illustrating this phenomenon is the function /: R

2 —>• R given by f(x

1,2:2) = 0 if xj • X'2 ^ 0 , while f(x 1,^2) — 1 if xi * x2 = 0 . This function is separately continuous at, (0,0) without being continuous at (0,0). On the other hand, if a function /: R'" —> R is continuous at A

>

0 then it is separately continuous at x° as well. In the paper [4] the author introduced the notion of strongly separately continuous function /: R'" —> R at and obtained the following result: A function /: R'

n —> R is continuous at a point x° if and only if it is strongly separately continuous at x° (see [4; Theorem 2.1]) In this paper we extend the notions of separately continuous function and strongly separately continuous function to the functions defined defined on the space i

1 and prove several basic results about functions. We show that there are essential differences between some properties of (strongly) separately continuous functions /: R

m —> R and the corresponding properties of functions /: P

2 —>• R. The paper consists of three sections. In the first section we introduce the notions of separately and strongly separately continuous function for the functions f:('

2 —* R and prove some basic results. In the second section we will investigate some properties of limit functions with respect to pointwise and weakly locally uniform convergence of sequences of (strongly) separately continuous functions /: f

2 —

y R and also with respect to pointwise convergence of transfinite sequences