Exercise 31, § 5

Let $f= (f_1, . . . , f_n):\mathbf{I}\rightarrow \mathbf{R}^n$ be a mapping of class $C^1$ such that $f'(t)= 0\not$ for all $t\in \mathbf{I}$.

(a) Assume that $f'_1(t)>0$ for all $t\in \mathbf{I}$. For all $j\in$ $\{2, . . . , n\}$, prove that there exists a continuous mapping $g_j:\mathbf{R}\rightarrow \mathbf{R}$ such that $f_j(x) =g_j(f_1(x))$ for all $x\in \mathbf{I}$. Deduce that $f(\mathbf{I})$ is isotopic to a segment. (b) Prove that there exists a real number $\delta >0$ and a continuous injective mapping $j: [-\delta ,1 +\delta ]\times \mathbf{B}_{n-1}\rightarrow \mathbf{R}^n$ such that $j(t,0) =f(t)$ for all $t\in \mathbf{I}$. Deduce from Exercise 30 that, for all $\tau \in [0,1[$, the set $f(\mathbf{I})$ is isotopic to $f([\tau ,1])$.

(c) Prove that $f(\mathbf{I})$ is isotopic to a segment.