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On Additive Almost Continuous Functions Under
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K. Ciesielski
Published/Copyright:
June 9, 2010
Abstract
We prove that the Covering Property Axiom 
, which holds in the iterated perfect set model, implies that there exists an additive discontinuous almost continuous function ƒ : ℝ → ℝ whose graph is of measure zero. We also show that, under , there exists a Hamel basis H for which, E+(H), the set of all linear combinations of elements from H with positive rational coefficients, is of measure zero. The existence of both of these examples follows from Martin's axiom, while it is unknown whether either of them can be constructed in ZFC.
As a tool for the constructions we will show that implies its seemingly stronger version, in which ω1-many games are played simultaneously.
Received: 2003-08-25
Revised: 2004-02-20
Published Online: 2010-06-09
Published in Print: 2005-December
© Heldermann Verlag
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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
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