On homeomorphisms and $C^{1}$ maps

Abstract

Our purpose in this article is first, following [8], to prove that if $α$, $β$ are any points of the open unit disc $D(0;1)$ in the complex plane ${\bf C}$ and $r$, $s$ are any positive real numbers such that ${\overline{D}}( α;r) \subseteq D(0;1)$ and ${\overline{D}}( β;s) \subseteq D(0;1)$, then there exist $t \in (0,1)$ and a homeomorphism $h : {\overline{D}}(0;1) \rightarrow {\overline{D}}(0;1)$ such that ${\overline{D}}( α;r) \subseteq D(0;t)$, ${\overline{D}}( β;s) \subseteq D(0;t)$, $h \left[ {\overline{D}}( α;r) \right] = {\overline{D}}( β;s)$ and $h = id$ on ${\overline{D}}(0;1) \setminus D(0;t)$, and second, following [9], to prove that if $q \in {\bf N} \setminus \{ 0, 1 \} $ and ${\bf B}({\bf 0};1)$ is the open unit ball in ${\bf R}^{q}$, while for any $t>0$, we set $f^{(t)}( {\bf x} ) = \frac{ t {\bf x} }{ 1 + (t-1) \Vert {\bf x} \Vert }$, whenever ${\bf x} \in {\overline{\bf B}}({\bf 0};1)$, then $f^{(t)} \rightarrow id$ in $C^{1} \left( {\overline{\bf B}}({\bf 0};1) , {\bf R}^{q} \right) $ as $t \rightarrow 1^{+}$.