Lemma 1

Let $f$ be a map from $I \times H$ into $E$ such that, on writing $f_x$ (for every $x \in H$) for the map $t \mapsto f(t, x)$ of $I$ into $E$, the following conditions are satisfied: 1° $f_x$ is regulated on $I$ for every $x \in H$; 2° the map $x \mapsto f_x$ of $H$ into the set $\mathscr{F}(I, E)$ of maps from $I$ into $E$ is continuous when one endows $\mathscr{F}(I, E)$ with the topology of compact convergence (Gen. Top., X, p. 278). Under these conditions:

1° For every continuous map $u$ of $I$ into $H$ the function $t \mapsto f(t, u(t))$ is regulated on $I$; more precisely, the right (resp. left) limit of this function at a point $t_0 \in I$ is equal to the right (resp. left) limit of the function $t \mapsto f(t, u(t_0))$ at the point $t_0$.

2° If $(u_n)$ is a sequence of maps of $I$ into $H$ which converges uniformly to a continuous function $u$ of $I$ into $H$ on every compact subset of $I$, then the sequence of functions $t \mapsto f(t, u_n(t))$ converges uniformly to $f(t, u(t))$ on every compact subset of $I$.

1 Let $c$ be the right limit of $f(t, u(t_0))$ at the point $t_0$; for every $\varepsilon > 0$ there is a compact neighbourhood $V$ of $t_0$ in $I$ such that $\|f(t, u(t_0)) - c\| \leq \varepsilon$ for $t \in V$ and $t > t_0$. On the other hand, there exists $\delta > 0$ such that the relations

imply $\|f(s, x) - f(s, u(t_0))\| \leq \varepsilon$ for all $s \in V$; if $W \subset V$ is a neighbourhood of $t_0$ in $I$ such that $\|u(t) - u(t_0)\| \leq \delta$ for every $t \in W$, then $\|f(t, u(t)) - c\| \leq 2\varepsilon$ for $t \in W$ and $t > t_0$, which proves that $c$ is the right limit of $f(t, u(t))$ at the point $t_0$.

2 Let $K$ be a compact subset of $I$; since $u$ is continuous on $I$, $u(K)$ is a compact subset of $H$; for every $\varepsilon > 0$ and $x \in u(K)$ there exists a number $\delta_x$ such that, for every $y \in H, \|y - x\| \leq \delta_x$ and every $t \in K$, one has $\|f(t, y) - f(t, x)\| \leq \varepsilon$. There is a finite number of points $x_i \in u(K)$ such that the closed balls with centre $x_i$ and radius $\frac{1}{2}\delta_{x_i}$ form a cover of $u(K)$; let $\delta = \operatorname{Min}(\delta_{x_i})$. By hypothesis there exists an integer $n_0$ such that for $n \geq n_0$ one has $\|u_n(t) - u(t)\| \leq \frac{1}{2}\delta$ for every $t \in K$. Now, for every $t \in K$ there exists an index $i$ such that

consequently one has $\|u_n(t) - x_i\| \leq \delta_x$, whence $\|f(t, u_n(t)) - f(t, u(t))\| \leq 2\varepsilon$ for every $t \in K$ and every $n \geq n_0$

For the rest of this section I will denote an interval contained in $\mathbf{R}$, not reducing to a single point, $H$ an open nonempty set contained in the normed space $E$, and $f$ a map from $I \times H$ into $E$ satisfying the hypotheses of lemma 1.