On Whitney pairs

Marianna Csörnyei

Fundamenta Mathematicae (1999)

  • Volume: 160, Issue: 1, page 63-79
  • ISSN: 0016-2736

Abstract

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A simple arc ϕ is said to be a Whitney arc if there exists a non-constant function f such that    l i m x ↦ x 0 ( | f ( x ) - f ( x 0 ) | ) / ( | ϕ ( x ) - ϕ ( x 0 ) | ) = 0 for every x 0 . G. Petruska raised the question whether there exists a simple arc ϕ for which every subarc is a Whitney arc, but for which there is no parametrization satisfying    l i m t ↦ t 0 ( | t - t 0 | ) / ( | ϕ ( t ) - ϕ ( t 0 ) | ) = 0 . We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.

How to cite

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Csörnyei, Marianna. "On Whitney pairs." Fundamenta Mathematicae 160.1 (1999): 63-79. <http://eudml.org/doc/212381>.

@article{Csörnyei1999,
abstract = {A simple arc ϕ is said to be a Whitney arc if there exists a non-constant function f such that   $lim_\{x ↦ x_0\} (|f(x)-f(x_0)|)/(|ϕ(x)-ϕ(x_0)|) = 0$ for every $x_0$. G. Petruska raised the question whether there exists a simple arc ϕ for which every subarc is a Whitney arc, but for which there is no parametrization satisfying   $lim_\{t ↦ t_0\} (|t-t_0|)/(|ϕ(t)-ϕ(t_0)|) = 0$. We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.},
author = {Csörnyei, Marianna},
journal = {Fundamenta Mathematicae},
keywords = { function; strictly monotone Whitney arc; Whitney pair},
language = {eng},
number = {1},
pages = {63-79},
title = {On Whitney pairs},
url = {http://eudml.org/doc/212381},
volume = {160},
year = {1999},
}

TY - JOUR
AU - Csörnyei, Marianna
TI - On Whitney pairs
JO - Fundamenta Mathematicae
PY - 1999
VL - 160
IS - 1
SP - 63
EP - 79
AB - A simple arc ϕ is said to be a Whitney arc if there exists a non-constant function f such that   $lim_{x ↦ x_0} (|f(x)-f(x_0)|)/(|ϕ(x)-ϕ(x_0)|) = 0$ for every $x_0$. G. Petruska raised the question whether there exists a simple arc ϕ for which every subarc is a Whitney arc, but for which there is no parametrization satisfying   $lim_{t ↦ t_0} (|t-t_0|)/(|ϕ(t)-ϕ(t_0)|) = 0$. We answer this question partially, and study the structural properties of possible monotone, strictly monotone and VBG* functions f and associated Whitney arcs.
LA - eng
KW - function; strictly monotone Whitney arc; Whitney pair
UR - http://eudml.org/doc/212381
ER -

References

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  1. [1] A. M. Bruckner, Creating differentiability and destroying derivatives, Amer. Math. Monthly 85 (1978), 554-562. Zbl0403.26002
  2. [2] A. M. Bruckner, Differentiation of Real Functions, CRM Monograph Ser. 5, Amer. Math. Soc., Providence, 1994, pp. 88-89. 
  3. [3] M. Laczkovich and G. Petruska, Whitney sets and sets of constancy, Real Anal. Exchange 10 (1984-85), 313-323. Zbl0593.26007
  4. [4] S. Saks, Theory of the Integral, Dover Publ., New York, 1964, pp. 228-240. 
  5. [5] H. Whitney, A function not constant on a connected set of critical points, Duke Math. J. 1 (1935), 514-517. Zbl0013.05801

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