Exercise 6, § 3

Let $I = [a, b]$, $A = [c, d]$ be two compact intervals on $\mathbf{R}$; let $f$ be a function defined on $I \times A$, with values in a complete normed space $E$ over $\mathbf{R}$, such that for all $\alpha \in A$, the map $t \mapsto f(t, \alpha)$ is regulated on $I$, that $f$ is bounded on $I \times A$, and that the set $D$ of points of discontinuity of $f$ in $I \times A$ is met in a finite number of points by each line $x = x_0$ and each line $\alpha = \alpha_0$ ($x_0 \in I, \alpha_0 \in A$).

a) Show that the function $g(\alpha) = \int_a^b f(t, \alpha)\,dt$ is continuous on $A$ (given $\alpha_0 \in A$ and $\varepsilon > 0$, show that there is a neighbourhood $V$ of $\alpha_0$ and a finite number of intervals $J_k$ contained in $I$ with the sum of their lengths $\leq \varepsilon$, such that, if $J$ denotes the complement in $I$ of $\bigcup_k J_k$, then $f$ is continuous on $J \times V$).

b) Show that the formula for interchanging the order of integration (II, p. 77, formula (15)) is still valid (same method as in $a$) ).