Solution
= Solution
\b[No.] At $a=0$, let
$$
f(x)=\sqrt[3]x,\qquad g(y)=y^3.
$$
The <function> $g$ is <differentiable> at $f(0)=0$, and
$$
(g\circ f)(x)=x,
$$
so the composite has <derivative> one at zero. But
$$
\frac{f(h)-f(0)}h=|h|^{-2/3}
$$
is unbounded, so $f$ is not <differentiable>. This is a <nondifferentiable inner function with a differentiable nondegenerate composite>.