On compactness and connectedness of the paratingent | Zendy

Wojciech Zygmunt | Zendy

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On compactness and connectedness of the paratingent

Author(s) -

Wojciech Zygmunt

Publication year - 2016

Publication title -

annales universitatis mariae curie-sklodowska sectio a – mathematica

Language(s) - English

Resource type - Journals

eISSN - 2083-7402

pISSN - 0365-1029

DOI - 10.17951/a.2016.70.2.91

Subject(s) - compact space , social connectedness , mathematics , neighbourhood (mathematics) , lipschitz continuity , combinatorics , function (biology) , set (abstract data type) , pure mathematics , discrete mathematics , mathematical analysis , computer science , psychology , evolutionary biology , psychotherapist , programming language , biology

In this note we shall prove that for a continuous function \(\varphi : \Delta\to\mathbb{R}^n\), 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.

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