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

On separately continuous functions /:£

2>-R 17 Theorem 3.1. There exists a strongly separately continuous function h: e R and a residual (and, consequently, dense) set E in f'

2 such that h(x) = 0 for all x £ E and h(y) ^ 0 for some y £ t

2 \ E. oo Proof. Denote by H the set of all x = (xj)jL

J £ C

2 for which ^ xj converges. Put. j=

1 oo E - I

2 \ H and define h: i

2 R by h{x) = xj for a11 x ^ 11 and hi

x) = 0 J'=I otherwise. According to [7; Theorem 3.1.] (it suffices to put o

n — 1 for all n = 1, 2,... and p = q = 2) the set E is residual in C

2. To complete the proof it. suffices to show that h is strongly separately continuous on C

2

. Let x° = )Ji

1 G Í

2 and k £ N. We show that h is strongly separately continuous at x° with respect, to xA-. Let £ > 0. If £ — (xj)JL

1 G B(x°,e), then also x = (xi,

..., x

k+i ) G oo ( B(x°,e). If x G H and h(x) = Y, X

j, then \h(x)-h{x )| - |ar

A-xJ| < ß{x,x°) < e. 3 = 1 If x i H, then h(x) = h{x) = 0 and we have \h{x) - h(x') \ = 0 < e. This yields that h is strongly separately continuous at x° with respect to x^. In connection with determining sets for strongly separately continuous functions on i

2 the following observation seems to be useful. Let M be a subset of Í

2 and M is the set of all y — (yj)JL

1 G t

2 such that there exists x = (xj)Ji

1 G M for which the set {j G N : xj / yj} is finite. It is obvious, that M C A/, M — M and M is a set of type (Pj). Similarly to the proof of Theorem 1.4 it can be checked that for any subset M C t

2 the function g: t

2 —

Y R given by g(x) — 0 for all x £ M and ^(x) = 1 otherwise is strongly separately continuous. Hence, we obtain: Proposition 3.2. If M is a subset of t

2 such that M ± C

2, th en M is not a determining set for the class of all (strongly) separately continuous functions on C . It. is easy to see that if M C ('

2 and card M < c, c being the cardinality of continuum, then M ^ (

2 (evidently, there exists y = {yj)j

(L

1 G f

2 such that for each x - (xj

)Ji

, G M, {j £ N : Xj — yj} — 0). Hence, as a consequence of Proposition 3.2 we obtain. Proposition 3.3. If M C £

2 is a determining set for the class of all (strongly

) separately continuous functions on f

2, then card M — c. References [1] BRUCKNER, A. M., Differentiation of Real Functions

, Spinger-Verlag, BerlinHeidelberg-New York, 1978. [2] CARROL, F. M., Separately continuous functions, Am er. Math. Monthly 78 (1971), 175.