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On the Convergence of Sequences of Functions which are Discontinuous on Countable Sets
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Z. Grande
Published/Copyright:
June 9, 2010
Abstract
Let (X, TX) be a topological space and let (Y, dY) be a metric space. For a function ƒ : X → Y denote by C(ƒ) the set of all continuity points of ƒ and by D(ƒ) = X\C(ƒ) the set of all discontinuity points of ƒ. Let
C(X, Y) = {ƒ : X → Y; ƒ is continuous},
H(X, Y) = {ƒ : X → Y; D(ƒ) is countable},
H1( X, Y) = {ƒ : X → Y; ∃h∈C(X, Y) {x; ƒ(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).
Key words and phrases.: Baire 1 class; density topology; quasiuniform convergence; discrete convergence; transfinite sequences
Received: 2003-11-12
Revised: 2004-03-17
Published Online: 2010-06-09
Published in Print: 2005-December
© Heldermann Verlag
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Keywords for this article
Baire 1 class;
density topology;
quasiuniform convergence;
discrete convergence;
transfinite sequences
Articles in the same Issue
- On Additive Almost Continuous Functions Under
- Maximal Solutions and Existence Theory for Fuzzy Differential and Integral Equations
- Projections in Weakly Compactly Generated Banach Spaces and Chang's Conjecture
- Dominated Convergence and Stone-Weierstrass Theorem
- Optimality Conditions and Duality for Multiobjective Control Problems
- On the Convergence of Sequences of Functions which are Discontinuous on Countable Sets
- Euler-Poincaré Formalism of Coupled KdV Type Systems and Diffeomorphism Group on S1
- Stability Analysis and Comparison of the Models for Carcinogenesis Mutations in the Case of Two Stages of Mutations
