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

16 «T. Cincura, T. Salát, T. Visnyai Theorem 2.4. Let (f^: í

2 —> R)^^ be a transfínite sequence of functions which converges pointwise to a function f:£

2 —

> R on I'

2. If for all £ < Q the function ft is (strongly) separately continuous at x°, then the function f is also (strongly) separately continuous at x°. Proof. Let for each £ < the function be strongly separately continuous at a;

0 with respect to x^. We show that f is strongly separately continuous at x° with respect to x^. Let (a^'J^Lj be a sequence in £

2 which converges to a?

0, = (®j

n

))Jii. For each n 6 N put x(

n>' = (x<

n

),

..., 4+i> • • •)• suffices to check that lim (/(x

(n

)) - /(«c

(n

)')) = 0. Let n G N. For every ( < Q we have n—too liin (fs(

xin

)) - fd

x{n

Y)) = Sinc

e ft -> / on £2 w

e obtain /

e(z

(n

)) -> f{x<">) and fzixW) /(

x(»)'). Then there exists < fi such that f^

n

)) = /{x^) and ft(x(

n) ) = f(x^ ) holds for all £ > We can choose £

0 < ß such that for all n G N we have £

n < £„• Then for all n G N /

i

o(a?(

n>) = /(x

,(n

)) and = flxW). Clearly, lim (f(x^)-/(art»)')) = lim {ffJx^) -/<_ (ar<

n)')) = 0. Hence, n —oo n—too the function / is strongly separately continuous at with respect to xk. The case of separate continuity immediately follows from the known fact that a limit of a transfinite sequence^: R —> of continuous functions is a continuous function (see e. g. [10], [9]). 3. Determining sets for separately continuous functions f:£

2 —>• R If T is a class of (real) functions defined on a set X and M C X

, then the set M is said to be a determining set for T provided that any functions f,gET satisfying J\m = g\\i are coincidental on X. For the class Q of all separately continuous function of two variables the following result was proved (see [13], [11], [8]). Theorem B. Let Q be the class of all separately continuous functions defined on R

2. Then a set M C R

2 is a determining set for the class Q if and only if M is dense in R

2. Obviously, this result can be extended to the class of all separately continuous functions defined on R

m,m > 2. On the other hand, from Theorem 1.4 it follows that there exist dense subsets of the space £

2, e. g. £

2 \ S, S ,£

2\S where S, S are presented in Example 1.3, that are not determining sets for the class of all (strongly) separately continuous functions on £

2. Another example is given in the next theorem.