Exercise 4, § 3

Let $I$ be an interval of $\mathbf{R}$, $A$ a neighbourhood of a point $\alpha_0$ in the field $\mathbf{R}$ (resp. the field $\mathbf{C}$), and $f$ a continuous map of $I \times A$ into a complete normed space $E$ over $\mathbf{R}$, such that $f'_\alpha(x, \alpha)$ exists and is continuous on $I \times A$. Let $a(\alpha)$, $b(\alpha)$ be two continuous functions defined on $A$, with values in $I$, such that one has identically $f(a(\alpha), \alpha) = f(b(\alpha), \alpha) = 0$ on $A$. Show that the function $g(\alpha) = \int_{a(\alpha)}^{b(\alpha)} f(t, \alpha)\,dt$ admits a derivative equal to $\int_{a(\alpha_0)}^{b(\alpha_0)} f'_\alpha(t, \alpha_0)\,dt$ at the point $\alpha_0$, even if $a$ and $b$ are not differentiable at the point $\alpha_0$ (let $M$ be the supremum of $\|f'_\alpha(x, \alpha)\|$ on a compact neighbourhood of $(b(\alpha_0), \alpha_0)$; note, applying Bolzano's theorem to $b(\alpha)$, that for every $x$ belonging to the interval with endpoints $b(\alpha_0)$ and $b(\alpha)$, one has, for $\alpha$ sufficiently close to $\alpha_0$, that $\|f(x, \alpha)\| \leq M |\alpha - \alpha_0|$).