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

14 «T

. Cincura, T. Salát, T. Visnyai every neighbourhood U of we have U fl S ^ 0

, U n (£

2 \ S) ± 0, and this yields that g is discontinuous at On the other hand, let k £ N and = (, x = (xj)JL

1

: x = (

Xj)j

cL

i be arbitrary points of £

2 such that for all j / k

, Xj = xand x

£ = x

k. It is obvious that if x £ S, then also x E S and if x S

, then also x £ S. Hence we always obtain |#(x) — g(x )| = 0 so that for each x° £ £

2 and each k £ N the function g is strongly separately continuous at with respect to xjf. Remark. While all separately continuous functions /: R

m —» R belong to the first Bai re class B\, Theorem 1.4 shows that neither strongly separately continuous nor separately continuous functions /: £

2 —» R have this property. The function g: Í

2 —

> R defined in the proof of Theorem 1.4 does not belong to Bi because the set of all discontinuity points of g is a set of the second Baire category. We close this section with two examples. The function f:£

2 —

> R define oo by f{xi,a;2) •••) — I if Y1 xk € Q, Q being the set of all rationals, and k = 1 f(xi, X2i • •.) = 0 otherwise is an example of a function which is nowhere separately continuous. The function g:£

2 —y R given by g(xi, ,...) = 0 if x\ • x,2 ^ 0 while g(xi, X2,

...) = 1 in the opposite case is separately continuous at (0, U,

...) without being strongly separately continuous at this point. 2. Limit functions of sequences of separately continuous functions /: t

2 R If a sequence {f

n-.£

2 converges pointwise to a. function /:i

2 —» R and all f

n are (strongly) separately continuous, then the function / need not be separately continuous. Theorem 2.1. There exists a sequence (f

n: £

2 —» R)^!, of functions each of which is continuous on C'

2 such that it converges pointwise to a function f:£

2—t R which is not separately continuous on £

2. Proof. For each n £ N define a function g

n: R —> R by g

n(x) = sin ^ for all x £ (

(

n +'

)^

n

, i) and g

n{x) = 0 otherwise. It is clear that all g

n are continuous functions on R and the sequence (g

n)'^L

1 converges pointwise to the function g: R -» R given by g(x) = sin ± for all x £ (Ü, £) and g(x) = Ü otherwise. Obviously, g is discontinuous at 0. For each n £ N define a function f

n:£

2 —> R by f

n(xi, X2,

...) = g

n(xj

) and let /: £

2 —

> R be the function given by X2,

...) = g(xi). It is evident that for all n £ N, f

n is a continuous function on £

2 (f

n = g

n°Pi

, where p\\í

2 —

> R is the first projection) and / is not separately continuous at the point (0, 0,...) with respect to X\. Clearly, the sequence (f

n)£°

=

1 converges pointwise to /.