Exercise 3, § 3

Let $f(x, \alpha) = 1/\sqrt{1 - 2\alpha x + \alpha^2}$ for $-1 < x < +1$ and $\alpha \in \mathbf{R}$; show that the function $g(\alpha) = \int_{-1}^{+1} dx/\sqrt{1 - 2\alpha x + \alpha^2}$ is continuous on $\mathbf{R}$, but has no derivative at $\alpha = 1$ and $\alpha = -1$; show that $f'_\alpha(x, \alpha)$ exists for all $\alpha \in \mathbf{R}$ and for $x \in I = ]-1, +1[$, and is continuous on $I \times \mathbf{R}$, and that the integral $\int_{-1}^{+1} f'_\alpha(x, \alpha)\,dx$ exists for all $\alpha \in \mathbf{R}$, but verify that this integral is not uniformly convergent on a neighbourhood of the point $\alpha = 1$ or of the point $\alpha = -1$.