Regularity of a power times a function with a simple zero ($F(x)=|x{|}^p\psi (x),F(0)=0$)

Suppose $\psi$ is differentiable at zero, $\psi (0)=0$ and ${\psi }^{\prime }(0)=a\ne 0$. Since $\psi (x)/x\to a$, the displayed function satisfies $F(x)\sim a\mathrm{sgn}(x)|x{|}^{p+1}$ 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 $p\ge 0$: 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.