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 13 From the above definition it follows the following: Proposition 1.2. (a

) If a function /: t'

2 —> R is continuous at x°, then f is strongly separately continuous at x°. (b) If a function f:i

2 —

> R is strongly separately continuous at x°, then fis separately continuous at x°. Proof, (a) Let (íc^

íJ

^

J be a. sequence in £

2 which converges to = (xS"

))?i

1. Then, obviously, lim /(ar<

n>) = f(x°). Let k G N. For every n G N put •> n—• oo x(n)' = (xj"),...,Since q(x^',x°) < g(x°,x<

n>) for all n G N we obtain that lim x^' = J;

0 and it follows that lim f(x

{n

)') = f{x°). Hence, 11—I OO Tl—tOO lim (/(•£'"') — )) = 0 and this yields that fis strongly separately continuous 71 —

tOO at x° with respect to xk for arbitrary k E N. (b) Similarly to (a). In the paper [4] the following result was proved. Theorem A. A function f

: R

m —>• R is continuous at x° if and only if f is strongly separately continuous at x°. In the case of functions /: t

2 R only the implication presented in Proposition 1.2 (a) is valid and we show that there exist, strongly separately continuous functions /: £

2 —t R (on £

2) which are discontinuous at every point of the space f

2. F or defining such functions the following notion seems to be useful. A subset £ is said to be a. set of type (Pi) provided the following holds: If x = (xjG S

, y = (y.jjj'Li £ ^

2 a

nd {j € N; Xj ^ Vj) contains at most one element, then y G S. Next we present some examples of subsets S C £

2 such that is a set of type(Pi

) and S as well as £

2 \ S are dense in f

2. Example 1.3. (a) c> = {j? = (xj)JL

l G f

2'-j G N; xj is a rational (irrational, algebraic, transcendent) number} is a finite set (see [14]). (b) S' = j* = (xj)^ G £

2 : Exj < +oo j Theorem 1.4. There exists a function y: £

2 —> R such that y is strongly separately continuous on i

2 and g is discontinuous at every point of £

2. Proof. Let S C (

2 be a set of type (Pi) such that S and f

2 \S are dense in I

2 (we can take some of the sets from Examples 1.3). Let CGR,C / 0. Define a function y:£

2 —y R by y(x) — c for all x G S and y(x) = 0 otherwise. If x° G ^

2, then for