Open AccessOn compactness and connectedness of the paratingent
Author(s) -
Wojciech Zygmunt
Publication year - 2016
Publication title -
annales universitatis mariae curie-skłodowska. 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.