Exercise 7, § 1

Let $\varphi$ be the canonical mapping of $\mathbf{R}$ onto $T = \mathbf{R}/\mathbf{Z}$, and let $f$ be the mapping $x \to \varphi(3x)$, restricted to the neighbourhood $V = [-1/8, +1/8]$ of $0$ in $\mathbf{R}$. Show that $f$ is a homeomorphism of $V$ onto $f(V)$ and satisfies condition 1) of Definition 2 of no. 3, but is not a local isomorphism from $\mathbf{R}$ to $T$. \*