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On the convergence of sequences of functions which are discontinuous on countable sets

Grande Z., Strońska E.

Journal of Applied Analysis

2005

Vol. 11, nr 2

247-259

Let (X, Tx) be a topological space and let (Y, dy) be a metric space. For a function f : X → y denote by C(f) the set of all continuity points of f and by D(f) = X\C(f) the set of all discontinuity points of f. Let C(X,Y) = {f : X → Y; f is continuous}, H(X, Y) = {f: X →Y; D{f) is countable}, H1(X, Y) = {f: X → Y; ∃h ∈c(x,Y) {x; f(x) ≠ h{x)} is countable}, and H2(X, Y) = H(X, Y) ∩ H1(X, Y). In this article we investigate some convergences (pointwise, uniform, quasiuniform, discrete and transfinite) of sequences of functions from H(X, Y), H1(X, Y) and H2(X, Y).

On the sets of continuity points of Darboux quasicontinuous functions

Grande M.

Demonstratio Mathematica

2003

Vol. 36, nr 4

819--825

We show that for every dense G-set A C R there is a Baire 1 Darboux bilaterally quasicontinuous function f : R-R such that C(f) = A, where C(f) denotes the set of all continuity points of f. Moreover we prove that for each G-set A there is a Baire 2 Darboux function f : R-R such that C(f) = A.