Exercise 13, § 6

Let $K$ be a compact subset of $\mathbf{R}^2$. For each real number $y$, let $\delta_1(K, y)$ [resp. $\delta_2(K, y)$] denote the diameter of $K \cap ([0, +\infty[ \times \{y\})$ [resp. $K \cap ([-\infty, 0] \times \{y\})$]. Show that the functions $y \to \delta_1(K, y)$ and $y \to \delta_2(K, y)$ are upper semi-continuous. Let $\varphi$ be an increasing continuous mapping of $[0, +\infty]$ into itself such that $\varphi(0) = 1$ and $\varphi(+\infty) = 2$, and let

Show that $f$ satisfies the conditions (CA_I) and (CA_{III}) and that we have $f(K \cup K') \leq f(K) + f(K')$ for each pair of compact subsets $K, K'$ of $\mathbf{R}^2$. But if $A$ is the closed set defined by $x \geq 0$ and $0 \leq y \leq 1$, show that $f_*(A) = 1$ and $f^*(A) = 2$. \*