Proposition 5

Suppose that the partial derivative $\mathbf{f}'_{\alpha}(x, \alpha)$ exists for all $x \in I$ and all $\alpha$ in an open neighbourhood $V$ of $\alpha_0$, and that, for all $\alpha \in V$, the map $x \mapsto \mathbf{f}'_{\alpha}(x, \alpha)$ is regulated on $I$. Under these conditions, if $x \mapsto \mathbf{f}'_{\alpha}(x, \alpha)$ converges uniformly on $I$ to $x \mapsto \mathbf{f}'_{\alpha}(x, \alpha_0)$ as $\alpha$ tends to $\alpha_0$, then the function $g(\alpha) = \int_a^b \mathbf{f}(t, \alpha)\,dt$ admits a derivative, given by the formula (12), at the point $\alpha_0$.

Indeed, for every $\varepsilon > 0$ there exists by hypothesis an $r > 0$ such that $|\alpha - \alpha_0| \leq r$ implies $\| \mathbf{f}'_{\alpha}(x, \alpha) - \mathbf{f}'_{\alpha}(x, \alpha_0) \| \leq \varepsilon$ for any $x \in I$. By props. 3 and 5 of I, p. 17 one has, for $|\alpha - \alpha_0| \leq r$ ($\alpha \neq \alpha_0$) and for all $x \in I$

which proves the uniform convergence of $\frac{\mathbf{f}(x, \alpha) - \mathbf{f}(x, \alpha_0)}{\alpha - \alpha_0}$ to $\mathbf{f}'_{\alpha}(x, \alpha_0)$ on I as $\alpha$ tends to $\alpha_0$ (remaining $\neq \alpha_0$), and so establishes formula (12).