Exercise 14, § 3

Let $\Gamma$ be the group of homeomorphisms of the numerical plane $\mathbf{R}^2$, endowed with the topology of simple convergence. Now prove that the mapping $(u, v) \mapsto v \circ u$ from $\Gamma \times \Gamma$ into $\Gamma$ is not continuous. (On the one hand consider a sequence of homeomorphisms $(v_n)$ such that $v_n$ leaves invariant every point $(x, y)$ for which $y$ does not belong to the interval $[1/(n+1), 1/n]$, and that the restriction of $v_n$ to the right $y = 2/(2n+1)$ is the translation $(x, y) \mapsto (x+1, y)$; on the other hand, consider the sequence $(u_n)$ of translations