A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that for every x ∈ γ; γ is 1-critical if there exists an f ∈ C¹(ℝⁿ) such that f’(x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in ℝ² each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This answers completely a problem of G. Petruska which was solved for n ≥ 3 by the first author in 1999.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-2-2,
author = {Marianna Cs\"ornyei and Jan Kali\v s and Lud\v ek Zaj\'\i \v cek},
title = {Whitney arcs and 1-critical arcs},
journal = {Fundamenta Mathematicae},
volume = {201},
year = {2008},
pages = {119-130},
zbl = {1152.26009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-2-2}
}
Marianna Csörnyei; Jan Kališ; Luděk Zajíček. Whitney arcs and 1-critical arcs. Fundamenta Mathematicae, Tome 201 (2008) pp. 119-130. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm199-2-2/