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1
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Quotients of peripherally continuous functions

100%
Kosman J.
Open Mathematics
|
2011
|
tom 9
|
nr 4
765-771
EN
We characterize the family of quotients of peripherally continuous functions. Moreover, we study cardinal invariants related to quotients in the case of peripherally continuous functions and the complement of this family.
2
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On the almost monotone convergence of sequences of continuous functions

76%
Grande Z.
Open Mathematics
|
2011
|
tom 9
|
nr 4
772-777
EN
A sequence (f n)n of functions f n: X → ℝ almost decreases (increases) to a function f: X → ℝ if it pointwise converges to f and for each point x ∈ X there is a positive integer n(x) such that f n+1(x) ≤ f n (x) (f n+1(x) ≥ f n(x)) for n ≥ n(x). In this article I investigate this convergence in some families of continuous functions.
3
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Points of continuity and quasicontinuity

76%
Borsík J.
Open Mathematics
|
2010
|
tom 8
|
nr 1
179-190
EN
Let C(f), Q(f), E(f) and A(f) be the sets of all continuity, quasicontinuity, upper and lower quasicontinuity and cliquishness points of a real function f: X → ℝ, respectively. The triplets (C(f),Q(f),A(f)), (C(f),E(f),A(f) and (Q(f),E(f),A(f)are characterized for functions defined on Baire metric spaces without isolated points.
4
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On some properties of Hamel bases and their applications to Marczewski measurable functions

64%
Dorais F., Filipów R., Natkaniec T.
Open Mathematics
|
2013
|
tom 11
|
nr 3
487-508
EN
We introduce new properties of Hamel bases. We show that it is consistent with ZFC that such Hamel bases exist. Under the assumption that there exists a Hamel basis with one of these properties we construct a discontinuous and additive function that is Marczewski measurable. Moreover, we show that such a function can additionally have the intermediate value property (and even be an extendable function). Finally, we examine sums and limits of such functions.
5
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A hierarchy in the family of real surjective functions

64%
Fenoy-Muñoz M., Gámez-Merino J. L., Muñoz-Fernández G. A., Sáez-Maestro E.
Open Mathematics
|
2017
|
tom 15
|
nr 1
486-501
EN
This expository paper focuses on the study of extreme surjective functions in ℝℝ. We present several different types of extreme surjectivity by providing examples and crucial properties. These examples help us to establish a hierarchy within the different classes of surjectivity we deal with. The classes presented here are: everywhere surjective functions, strongly everywhere surjective functions, κ-everywhere surjective functions, perfectly everywhere surjective functions and Jones functions. The algebraic structure of the sets of surjective functions we show here is studied using the concept of lineability. In the final sections of this work we also reveal unexpected connections between the different degrees of extreme surjectivity given above and other interesting sets of functions such as the space of additive mappings, the class of mappings with a dense graph, the class of Darboux functions and the class of Sierpiński-Zygmund functions in ℝℝ.
6
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On similarity between topologies

52%
Bartoszewicz A., Filipczak M., Kowalski A., Terepeta M.
Open Mathematics
|
2014
|
tom 12
|
nr 4
603-610
EN
Let T 1 and T 2 be topologies defined on the same set X and let us say that (X, T 1) and (X, T 2) are similar if the families of sets which have nonempty interior with respect to T 1 and T 2 coincide. The aim of the paper is to study how similar topologies are related with each other.
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