 Continuity and differentiability thresholds for a damped oscillation

Suppose $h$ is bounded by one in absolute value on a punctured neighbourhood of zero. Suppose also that there are positive sequences $p_n,q_n\to 0$ with $h(p_n)=1$ and $h(q_n)=0$. Extend $F_a(x)=|x{|}^{a}h(x)$ by $F_a(0)=0$. ThenWhen the derivative exists, it equals zero. For continuity, the bound $|F_a(x)|\le |x{|}^a$ gives the epsilon-delta definition of continuity with $\delta ={\epsilon }^{1/a}$ when $a>0$. If $a\le 0$, the values $F_a(p_n)=p_n^a$ remain at least one for sufficiently large $n$, contradicting continuity at zero.

For differentiability, the difference quotient is bounded in absolute value by $|x{|}^{a-1}$, so it tends to zero when $a>1$. Conversely its values along $q_n$ are zero, forcing any possible derivative to be zero. Its values along $p_n$ equal $p_n^{a-1}$ and fail to tend to zero when $a\le 1$.

For example, $h(x)=\sin (\pi \sin (1/x))$ has the required positive sequences $p_n=1/(2\pi n+\pi /6)$ and $q_n=1/(2\pi n)$. This separates the suppression needed for continuity from the stronger suppression needed for differentiability.