= Solution
Use the <derivative> definition and the specified value $f(0)=0$:
$$
\frac{f(h)-f(0)}h=A\omega|h|^p\frac{\sin(\omega h)}{\omega h}.
$$
The last factor tends to one. Therefore the limit is zero when $p>0$, equals $A\omega$ when $p=0$, and diverges to $+\infty$ when $p<0$. A finite <derivative> exists exactly when
$$
\boxed{p\ge0,\qquad f'(0)=\begin{cases}0,&p>0,\\A\omega,&p=0.\end{cases}}
$$
In particular $-1<p<0$ gives continuity without differentiability. The threshold is an example of <regularity of a power times a function with a simple zero>.
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