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

12 «T

. Cincura, T. Salát, T. Visnyai of (strongly) separately continuous functions /: £

2 —

> R. In the third section we will study determining sets for the class of (strongly) separately continuous functions on £

2. In this paper we, as usually, denote by t

2 the metric space consisting of all oo sequences x = (xj)!^ of real numbers such that xt < "t-0

0 endowed with the k-] metric g defined by Q(x,y) = - Vk)'' k = \ for all x,y E i

2

. If x° £ £

2 and S > 0, then B{x°,S) denotes the set {x E £

2 : g(x°,x) < 6}. 1. Separately and strongly separately continuous functions The definitions of separate and strong separate continuity of functions /: R

m —» R can be in a natural way extended to the case of functions /: £

2 —» R. Definition 1.1. (a) A function f: £

2 —» R is said to be separately continuous at a point x° = [x

<j)JL

1 E £

2 with respect to a variable xk provided that the function <pk- R- —> R defined by <Pk{t) = /(xj

1,

..., xjj^, i, x°

+

1,

...) is continuous at x°. If / is separately continuous at x° with respect to xk for all iGN, then / is said to be separately continuous at x°. If / is separately continuous at every point x° £ £

2, then / is said to be separately continuous on £

2. (b) A function f:£ 2 R is said to be strongly separately continuous at a point x° = (x°j)j

:L

1 £ £

2 with respect to a variable Xk provided that for each £ > 0 there exists J > 0 such that | f(x) — f(x') | < e holds for each x = (xj)Jl

1 £ B(x°,6), and x = (a?i,

..., x

k-\, x£, x

k+i,

...). If / is strongly separately continuous at x° with respect to x

k for all k £ N, then / is said to be strongly separately continuous at x°. The function /: £

2 —y R is said to be strongly separately continuous on £

2 provided that it is strongly separately continuous at every x

ü E £

2

. Remark. Observe that in Definition 1.1 (b) £(x°,x) < p(x°,x). Hence, if x E B(x°,S) , then x E B(x°

, <S) as well. It is also obvious that a function /: £

2 —>• R is strongly separately continuous at x° = (x

(j)JL

1 with respect to Xk if only if for any sequence in t

2 which converges to x° we obtain that lim (/(x^) — n—too f{x^')) = 0, where *<") = (a^)^ and x<">' = (x*"*,

..., xj^, • • •) fo

r all n E N.