= Regularity of a power times a function with a simple zero
{title2=$F(x)=|x|^p\psi(x),\quad F(0)=0$}
Suppose $\psi$ is differentiable at zero, $\psi(0)=0$ and $\psi'(0)=a\ne0$. Since $\psi(x)/x\to a$, the displayed function satisfies $F(x)\sim a\operatorname{sgn}(x)|x|^{p+1}$ near zero. It is a <continuous function> at zero exactly when $p>-1$. Its difference quotient is $|h|^p\psi(h)/h$, so it has a finite <derivative> there exactly when $p\ge0$: the value is $a$ for $p=0$ and zero for $p>0$. The nonzero-slope assumption is essential; a higher-order zero would change both thresholds.
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