Exercise 1, § 9

Let A be the compact subset of $\mathbf{R}$ consisting of 0 and the real numbers $1/n$, with $n$ an integer $\geq 1$. Show that when $\mathscr{C}^r(\mathbf{R};\mathbf{R})$ is given the topology of uniform $C^r$-convergence on A, the set of morphisms that are embeddings in the neighbourhood of A is not open in $\mathscr{C}^r(\mathbf{R};\mathbf{R})$ (consider a sequence of functions $(f_n)_{n\geq 1}$ such that $f_n(x) =x$ for $x\leq \frac{1}{n+1},f_n(x) =x-\frac{1}{n}$ for $x\geq \frac{1}{n}$).