Exercise 14, § 3

Let $\Gamma$ be the group of all homeomorphisms of the real plane $\mathbf{R}^2$, endowed with the topology of pointwise convergence. Show that the mapping $(u, v) \to v \circ u$ of $\Gamma \times \Gamma$ into $\Gamma$ is not continuous. [Consider a sequence of homeomorphisms $(v_n)$ such that $v_n$ leaves fixed every point $(x, y)$ for which $y \in [1/(n+1), 1/n]$, and such that the restriction of $v_n$ to the line $y = 2/(2n + 1)$ is the translation $(x, y) \to (x + 1, y)$; on the other hand, consider the sequence $(u_n)$ of translations