= Continuity and differentiability thresholds for a damped oscillation
{title2=$F_a(x)=|x|^ah(x),\quad F_a(0)=0$}
Suppose $h$ is bounded by one in <absolute value> on a punctured neighbourhood of zero. Suppose also that there are positive <sequences> $p_n,q_n\to0$ with $h(p_n)=1$ and $h(q_n)=0$. Extend $F_a(x)=|x|^ah(x)$ by $F_a(0)=0$. Then
$$
F_a\text{ is continuous at zero}\iff a>0,
\qquad
F_a\text{ is differentiable at zero}\iff a>1.
$$
When the <derivative> exists, it equals zero. For <continuity>, the bound $|F_a(x)|\leq|x|^a$ gives the <epsilon-delta definition of continuity> with $\delta=\varepsilon^{1/a}$ when $a>0$. If $a\leq0$, 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 $a\leq1$.
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>.
Back to article page
