Exercise 7, § 1

Show that the function equal to $\sin(1/x)$ for $x \neq 0$ and to 0 for $x = 0$, admits a strict primitive on $\mathbf{R}$ (remark that $x^2 \sin(1/x)$ admits a derivative at every point).

Deduce from this that if $g(x, u, v)$ is a polynomial in $u,\ v$, its coefficients being continuous functions of $x$ on an interval $I$ containing 0, then the function equal to $g(x, \sin 1/x, \cos 1/x)$ for $x \neq 0$, and to a suitable value $\alpha$ (to be determined) for $x = 0$, admits a strict primitive on $I$; give an example where $\alpha \neq g(0, 0, 0)$.*