# On Whitney pairs

Fundamenta Mathematicae (1999)

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

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topCsörnyei, 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

top- [1] A. M. Bruckner, Creating differentiability and destroying derivatives, Amer. Math. Monthly 85 (1978), 554-562. Zbl0403.26002
- [2] A. M. Bruckner, Differentiation of Real Functions, CRM Monograph Ser. 5, Amer. Math. Soc., Providence, 1994, pp. 88-89.
- [3] M. Laczkovich and G. Petruska, Whitney sets and sets of constancy, Real Anal. Exchange 10 (1984-85), 313-323. Zbl0593.26007
- [4] S. Saks, Theory of the Integral, Dover Publ., New York, 1964, pp. 228-240.
- [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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