κ-to-1 DARBOUX-LIKE FUNCTIONS

Ciesielski,; Gibson,; Natkaniec,

doi:10.2307/44153989

Cited by 3 publications

(8 citation statements)

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“…is easily seen to fulfill our requirements (see also [2], Proposition 1.15). Finally, the class φ(c) possesses also a continuous element.…”

Section: Definition 22 Introduce the Setmentioning

confidence: 53%

“…Indeed, it relies on the existence of continuous functions with a given assignment of the fibers. Many authors have investigated various aspects of this problem (see, e.g., [2,4,6,7,10] and references therein), giving answers to questions as "Does there exist a continuous k-to-1 function defined on a given topological space?". Actually, as we will see, the possibility to rearrange a map to a continuous one, depends only on the existence of a continuous function with a given assignment of the cardinality of the fibers.…”

Section: Introductionmentioning

confidence: 99%

“…This is a purely combinatorial classification of Y X . On the other hand, it discloses a variety of topological problems, which have been widely investigated in recent literature (see, for instance, [2,4,6,7,10]).…”

Section: Definition 22 Introduce the Setmentioning

confidence: 99%

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Rearrangeable Functions on the Real Line

Pata¹,

Ursino²

1998

Real Analysis Exchange

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“…is easily seen to fulfill our requirements (see also [2], Proposition 1.15). Finally, the class φ(c) possesses also a continuous element.…”

Section: Definition 22 Introduce the Setmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rearrangeable Functions on the Real Line

Pata¹,

Ursino²

1998

Real Analysis Exchange

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“…In Sections –, we discuss some aspects of the structure of k ‐to‐1 maps between graphs. For a recent paper on continuous 3‐to‐1 functions, not dealing with graphs, see .…”

Section: Introductionmentioning

confidence: 99%

“…We can, however, get an approximation of such a function by weakening the 3‐to‐1 condition. Following , given a nonnegative integer valued function

j : [0, 1] \to {double-struckZ}_{+}

, we say that a function f from [0, 1] onto [0, 1] is j ‐to‐1 if

| f^{- 1} (x) | = j (x)

for all

x \in [0, 1]

. We show that j ‐to‐1 functions from [0, 1] onto [0, 1] exist for functions

j : [0, 1] \to {double-struckZ}_{+}

such that

j (x) = 3

for

x \in (0, 1)

if and only if, in addition,

{j (0), j (1)} \subseteq {1, 2}

.…”

Section: Introductionmentioning

confidence: 99%

Wiggles and Finitely Discontinuous k‐to‐1 Functions Between Graphs

Gauci

Hilton

Stark

2013

Journal of Graph Theory

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A function between graphs is k‐to‐1 if each point in the codomain has precisely k preimages in the domain. In this article, we approach the topic of continuous, or finitely discontinuous, k‐to‐1 functions between graphs from three different points of view. Harrold (Duke Math J 5 (1939), 789–793) showed that there is no 2‐to‐1 continuous function from a closed interval onto a circle (i.e., from K2 onto C3). In the first part of this article, we describe all 3‐to‐1 continuous functions from an edge onto a cycle. Such a description is just one step away from a description of all 3‐to‐1 continuous functions from double-struckR onto double-struckR, which is in fact our main initial emphasis. Second, given two graphs, G and H, and an integer k≥1, and considering G and H as subsets of double-struckR3, Jo Heath gave a simple criterion for the existence of a finitely discontinuous k‐to‐1 function from G onto H. Such functions often involve a limiting construction which we call a wiggle. In the second part of this article, we give a simple formula (related to Jo Heath's construction) which counts the number of wiggles. The question of whether there is a continuous k‐to‐1 function (i.e., a k‐to‐1 map in the usual topological sense) from G onto H is more complicated. In the third part of this article, we consider complete graphs Kn and Km. In the cases where n and m have the same parity, and n≤m, then we determine exactly when there is a k‐to‐1 continuous function from Kn onto Km. Other cases are considered elsewhere (J. K. Dugdale, S. Fiorini, A. J. W. Hilton, J. B. Gauci, Discrete Math 310 (2010), 330–346 and J. B. Gauci, A. J. W. Hilton, J Graph Theory 65 (2010), 35–60).

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Continuous functions taking every value a given number of times

Kwiatkowska

2008

Acta Math Hung

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κ-to-1 DARBOUX-LIKE FUNCTIONS

Cited by 3 publications

References 0 publications

Rearrangeable Functions on the Real Line

Rearrangeable Functions on the Real Line

Wiggles and Finitely Discontinuous k‐to‐1 Functions Between Graphs

Continuous functions taking every value a given number of times

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