Exercise 5, § 3

Let $f$ be a continuous vector function on a compact interval $I = [0, a]$. Show that if, at the point $\alpha_0 \in I$, there is an $\varepsilon > 0$ such that $(f(x) - f(\alpha_0))/|x - \alpha_0|^{1/2 + \varepsilon}$ remains bounded as $x$ tends to $\alpha_0$, then the function $g(\alpha) = \int_0^\alpha f(x)\,dx/\sqrt{\alpha - x}$ admits a derivative equal to

at the point $\alpha_0$ for $\alpha_0 > 0$, and to 0 for $\alpha = 0$.

When $f$ is the real function $\sqrt{\alpha_0 - x}$, show that the function $g$ has an infinite derivative at the point $\alpha_0$.