Differentiability restored by a quadratic zero
= Differentiability restored by a quadratic zero
{title2=$g(t)=t^2f(t)$}
For a <continuous function> $f:\mathbb R\to\mathbb R$, $g(t)=t^2f(t)$ is <differentiable> at zero with $g'(0)=0$, because $[g(h)-g(0)]/h=hf(h)\to0$. At any nonzero point $x$, $g$ is <differentiable> exactly when $f$ is: one direction uses the <product rule>, and the converse applies it to $f(t)=t^{-2}g(t)$ near $x$.