On compactness and connectedness of the paratingent

Wojciech Zygmunt

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2016)

  • Volume: 70, Issue: 2
  • ISSN: 0365-1029

Abstract

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In this note we shall prove that for a continuous function ϕ : Δ → ℝ n , where Δ ⊂ ℝ ,  the paratingent of ϕ at a ∈ Δ is a non-empty and compact set in ℝ n if and only if ϕ satisfies Lipschitz condition in a neighbourhood of a . Moreover, in this case the paratingent is a connected set.

How to cite

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Wojciech Zygmunt. "On compactness and connectedness of the paratingent." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 70.2 (2016): null. <http://eudml.org/doc/289745>.

@article{WojciechZygmunt2016,
abstract = {In this note we shall prove that for a continuous function $\varphi : \Delta \rightarrow \mathbb \{R\}^n$, where $\Delta \subset \mathbb \{R\}$,  the paratingent of $\varphi $ at $a\in \Delta $ is a non-empty and compact set in $\mathbb \{R\}^n$ if and only if $\varphi $ satisfies Lipschitz condition in a neighbourhood of $a$. Moreover, in this case the paratingent is a connected set.},
author = {Wojciech Zygmunt},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {},
language = {eng},
number = {2},
pages = {null},
title = {On compactness and connectedness of the paratingent},
url = {http://eudml.org/doc/289745},
volume = {70},
year = {2016},
}

TY - JOUR
AU - Wojciech Zygmunt
TI - On compactness and connectedness of the paratingent
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2016
VL - 70
IS - 2
SP - null
AB - In this note we shall prove that for a continuous function $\varphi : \Delta \rightarrow \mathbb {R}^n$, where $\Delta \subset \mathbb {R}$,  the paratingent of $\varphi $ at $a\in \Delta $ is a non-empty and compact set in $\mathbb {R}^n$ if and only if $\varphi $ satisfies Lipschitz condition in a neighbourhood of $a$. Moreover, in this case the paratingent is a connected set.
LA - eng
KW -
UR - http://eudml.org/doc/289745
ER -

References

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  1. Aubin, J. P., Frankowska, H., Set-Valued Analysis, Birkhauser, Boston, Massachusetts, 1990. 
  2. Bielecki, A., Sur certaines conditions necessaires et suffisantes pour l’unicite des solutions des systemes d’equations differentielles ordinaires et des equations au paratingent, Ann. Univ. Mariae Curie-Skłodowska Sect. A 2 (1948), 49-106. 
  3. Bouligand, G., Introduction a la geometrie infinitesimale directe, Vuibert, Paris, 1932. 
  4. Choquet, G., Outils topologiques et metriques de l’analyse mathematique, Centre de Documentation Univ., Course redige par C. Mayer, Paris, 1969. 
  5. Fedor, M., Szyszkowska, J., Darboux properties of the paratingent, Ann. Univ. Mariae Curie-Skłodowska Sect. A 62 (2008), 67-74. 
  6. Mirica, S., The contingent and the paratingent as generalized derivatives for vectorvalued and set-valued mappings, Nonlinear Anal. 6 (1982), 1335-1368. 

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