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Use the derivative definition and the specified value $f(0)=0$: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 whenIn 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.