On homeomorphisms and $C^1$ maps

Abstract

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