Solution
= Solution
\b[No.] Take $a=0$ and
$$
f(x)=x,\qquad g(x)=1+|x|.
$$
Then $f$ is <differentiable> and
$$
(fg)(x)=x+x|x|.
$$
Since $x|x|/x=|x|\to0$, the product is <differentiable> at zero with
$$
(fg)'(0)=1\ne0.
$$
However, $g$ has one-sided <derivatives> $-1$ and $1$ and is not <differentiable> at zero. This is a <nondifferentiable factor with a differentiable nondegenerate product>.