On compactness and connectedness of the paratingent

Abstract

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.