On a property of functions. (English) Zbl 1107.26004

Grande, Marcin

Real Anal. Exch. 31(2005-2006), No. 2, 469-476 (2006).

A function \(f: X\to Y\), where \((X,\rho_X)\) and \((Y,\rho_Y)\) are metric spaces, is almost continuous, in the sense of J. Stallings [Fundam. Math. 47, 249–263 (1959; Zbl 0114.39102)], iff each set \(U\), open in the product space \(X\times Y\), that contains \(G(f)\), the graph of \(f\), also contains the graph of a continuous function \(g: X\to Y\). Similarly, \(f\) is termed graph continuous iff cl \(G(f)\) contains the graph, \(G(g)\), of a continuous function \(g: X\to Y\). The later concept was introduced by Z. Grande [Demonstr. Math. 11, 519–526 (1978; Zbl 0392.26002)] but with a different name.
In the present work, the author introduces a new concept of near continuity of \(f: X\to Y\) in the following manner: For each positive real number \(\eta\), let \(A_\eta(f)=\bigcup_{x\in X} (K(x,\eta)\times K(f(x),\eta))\), where \(K(t,\varepsilon)\) denotes the open ball with center \(t\) and radius \(\varepsilon\). Then \(f\) has property (a) iff for each positive \(\eta\), there is a continuous function \(g: X\to Y\) such that \(G(g)\subset A_\eta(f)\).
The author observes that the almost continuous functions and the graph continuous functions all have the property (a), but shows that there are functions with closed graphs and satisfying the condition (a) which are neither almost continuous nor graph continuous. Additionally the author establishes several other interesting results, relating to this concept; two of which are:
Theorem 2. Let \((X,\rho_X)\) and \((Y,\rho_Y)\) be metric spaces. If each of the functions \(f_n: X\to Y\) has property (a) and if \(f_n\rightrightarrows f\), then \(f\) has property (a).
Theorem 7. Let \((X,\rho_X)\) be a separable, dense in itself metric space, and let \((Y,\rho_Y)\) be a metric space. For each function \(f: X\to Y\) there is a sequence of functions \(\{f_n: X\to Y\}^\infty_{n=1}\) such that each \(f_n\) has property (a) and the sequence converges discretely to \(f\).

Reviewer: Robert E. Zink (Lafayette)

MSC:

26A15

Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Keywords:

almost continuity of Stallings; graph continuity; discrete convergence; uniform convergence; sum; product

Citations:

Zbl 0114.39102; Zbl 0392.26002