Let $q : [0,1] \rightarrow \mathbf{R}$ defined by
$$\left\{ \begin{array}{l}
q(x) = x \cos\left(\frac{\pi}{x}\right) \text{ for } x > 0 \\
q(0) = 0
\end{array} \right.$$
Show that for all $x_{0} \in [0,1]$, $\alpha_{q}(x_{0}) = 1$, but that $\frac{\omega_{q}(h)}{\sqrt{h}}$ does not tend to 0 when $h$ tends to 0.