Exercise 12, § 3

Let $X$ be a topological space, $Y$, $Z$ two uniform spaces, $\mathfrak{S}$ a set of subsets of $X$, $\mathfrak{L}$ a set of subsets of $Y$.

(a) Let $u_0 \in \mathscr{C}(X; Y)$. Prove that if, for every subset $A \in \mathfrak{S}$, $u_0(A)$ is contained in a set $B \in \mathscr{E}$, then the mapping $v \mapsto v \circ u_0$ from $\mathscr{C}_x(Y; Z)$ into $\mathscr{C}_e(X; Z)$ is continuous.

(b) Let $H$ be a subspace of $\mathscr{C}_e(X; Y)$, and let $v_0 \in \mathscr{C}(Y; Z)$. Prove that if, for every subset $A \in \mathfrak{S}$, $v_0$ is uniformly continuous in $H(A)$, the mapping $u \mapsto v_0 \circ u$ from $H$ into $\mathscr{C}_e(X; Z)$ is continuous.

(c) Let $u_0 \in \mathscr{C}(X; Y)$, $v_0 \in \mathscr{C}(Y; Z)$. Suppose that for every $A \in \mathfrak{S}$, there exists $B \in \mathscr{E}$ and an entourage $V$ of $Y$ such that: 1° $V(u_0(A)) \subset B$; 2° $v_0$ is uniformly continuous in $B$. Then, the mapping $(u, v) \mapsto v \circ u$ from $\mathscr{C}_e(X; Y) \times \mathscr{C}_x(Y; Z)$ into $\mathscr{C}_e(X; Z)$ is continuous at the point $(u_0, v_0)$. In particular, the mapping $(u, v) \mapsto v \circ u$ from $\mathscr{C}_u(X; Y) \times \mathscr{C}_u(Y; Z)$ into $\mathscr{C}(X; Z)$ is continuous at every point $(u_0, v_0)$ such that $v_0$ is uniformly continuous in $Y$.

(d) Let $v_0$ be the homeomorphism $x \mapsto x^3$ of $\mathbf{R}$ onto itself. Prove that the mapping $u \mapsto v_0 \circ u$ from $\mathscr{C}_u(\mathbf{R}; \mathbf{R})$ into itself is not continuous at certain points.