Exercise 17, § 6

Let $K$ be a compact subset of $\mathbf{R}^2$. For every real number $y$, denote by $\delta_1(K, y)$ and $\delta_2(K, y)$ the diameters of the respective intersections of $K$ and

Show that $y \mapsto \delta_1(K, y)$ and $y \mapsto \delta_2(K, y)$ are upper semi-continuous. Let $\varphi$ be an increasing and continuous mapping from $(0, +\infty)$ into itself such that $\varphi(0) = 1$ and $\varphi(+\infty) = 2$; put

Show that $f$ satisfies the conditions of Exercise 16 of IX, p. 122 and that one has

for every pair of compact subsets of $\mathbf{R}^2$. But if $A$ is the closed set defined by $x \geq 0, 0 \leq y \leq 1$ show that one has $f_*(A) = 1$ and $f^*(A) = 2$.