Exercise 3, § 3

Let $X$ be a metric space, $d$ the metric on $X$. For each $x \in X$ let $d_x$ be the real-valued function $y \to d(x, y)$ which is continuous on $X$. Show that the mapping $x \to d_x$ is an isometry of $X$ onto a subspace of $\mathscr{C}_u(X; \mathbf{R})$ [endowed with the pseudometric $\delta(u, v) = \sup_{x \in X} |u(x) - v(x)|$].