Proposition 7

Let I be an arbitrary interval in $\mathbf{R}$, and $f$ a continuous function on $I \times V$. Suppose that: 1. the partial derivative $f'_\alpha(x, \alpha)$ exists for all $x \in I$ and all $\alpha \in V$, and, for all $\alpha \in V$, the map $x \mapsto f'_\alpha(x, \alpha)$ is regulated on I; 2. for all $\alpha \in V$, $f'_\alpha(x, \beta)$ converges uniformly to $f'_\alpha(x, \alpha)$ on every compact interval contained in I, as $\beta$ tends to $\alpha$; 3. the integral $\int_I f'_\alpha(t, \alpha) \, dt$ is uniformly convergent on $V$; 4. the integral $\int_I f(t, \alpha_0) \, dt$ is convergent.

In these circumstances the integral $g(\alpha) = \int_I f(t, \alpha) \, dt$ is uniformly convergent on $V$, and the function $g$ admits at every point of $V$ a derivative given by the formula

The uniform convergence of $\int_I f'_\alpha(t, \alpha) \, dt$ on $V$ means that the function $\alpha \mapsto \int_J f'_\alpha(t, \alpha) \, dt$ converges uniformly on $V$ with respect to the filter of sections $\Phi$ of the directed set $\mathfrak{K}(I)$ of compact intervals $J$ contained in I. Let us put $u_J(\alpha) = \int_J f(t, \alpha) \, dt$; the hypotheses show that on the one hand $u_J(\alpha_0)$ has a limit with respect to $\Phi$, and on the other hand, by virtue of prop. 5 of II, p. 74, that $u'_J(\alpha) = \int_J f'_\alpha(t, \alpha) \, dt$ for all $\alpha \in V$. We can therefore apply th. 1 of II, p. 52 to the functions $u_J$, the rôle of the set of indices being taken here by $\mathfrak{K}(I)$, and that of the filter on this set by the filter $\Phi$; the proposition follows immediately.