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 15 It is natural to ask whether some of various types of convergence of functions which are stronger than the point wise convergence can guarantee that the limit function of a sequence of (strongly) separately continuous functions on f 2 with respect to this type of convergence is also a (strongly) separately continuous function on £ 2. Next we show that there is a weaker type of locally uniform convergence (see [14], [5; p. 149]) which fulfills this requirement in the case of strongly separately continuous functions on Í 2. Definition 2.2. Let X he a topological space, (f n: X R)^! be a sequence of functions and a' 0 £ A'. A sequence (/n)nLi ' s s ai (' to converge weakly locally uniformly to a function f: X —> R at x° if for every £ > 0 there exist S > 0 and p £ N such that \f n{%) — /(•*')! < e holds for each n £ N with n > p and each xG B{x°,S). If a sequence (f n )j converges weakly locally uniformly to a function / at every point x° £ A', then it is said to converge weakly locally uniformly to f on X. Theorem 2.3. If a sequence (f n'£ 2 R )j converges weakly locally uniformly to f: £ 2 —y R at x° £ f 2 and for each n £ N the function f n is strongly separately continuous at x°, then the function f is also strongly separately continuous at x° . Proof. Let k £ N. We will prove that / is strongly separately continuous at x° with respect to x k. bet £ > 0. Since converges weakly locally uniformly to / at x° there exist an open ball B(x°,Si) and p £ N such that \f n(x) — f(x)\ < - holds for all n > p and x £ B(x°, ái). The function f p is strongly separately continuous at x° with respect to x k and it follows that there exists 62 > 0 such that \f P[x) — f p (x )| < I holds for each x = (xj)J± 1 £ B(x°, Ó2) and x = (xj ,..., x k-\ , x£, x k +i, .. .). Put S = min{d"i, <5-2}• Then for each x £ B(x°,S) we obtain that |/ ;,(x) — f p(x )| < |/ p(x) — /(x)| < I and because o(x , x°) < f?(x°,x) < á we have also \f P(x ) — /(x')| < §. Hence, for all x G B(x°,S) we obtain |/(x) - f{x)\ < |/(x) - f p(x)\ + I fp (x ) - f p(x )| + I f p(x ) - /(x )| < £ and this yields that, / is strongly separately continuous at x with respect to x k. In the rest of this section we will investigate some properties of limit functions of convergent transfinite sequences of (strongly) separately continuous functions. Recall that a transfinite sequence is the first uncountable ordinal) in a metric space (A', rr) converges to a point x £ X ( we write x^ — > x) if for every £ > U there exists £0 < & such that cr(x£,x) < £ holds for each £0 < £ < £2. It is well known (see e.g. [9]) that if x^ —> x in a metric space (A', <r), then there exists £0 < Q such that X£ = x holds for each £ > A transfinite sequence (/f : M —¥ R)^<n of functions, M is a set, converges pointwise to a function /: M —» R (we write ft f) o n if for each x £ M we have /^(x) —y f(x) in R. In the next theorem we show that the pointwise convergence of transfinite sequences of functions preserves (strong) separate continuity.
