  We prove that all the composition operators $T_f(g):= f\circ g$, which take
the Adams-Frazier space $W^{m}_{p}\cap \dot{W}^{1}_{mp}(\mathbb{R}^n)$ to
itself, are continuous mappings from $W^{m}_{p}\cap
\dot{W}^{1}_{mp}(\mathbb{R}^n)$ to itself, for every integer $m\geq 2$ and
every real number $1\leq p<+\infty$. The same automatic continuity property
holds for Sobolev spaces $W^m_p(\mathbb{R}^n)$ for $m\geq 2$ and $1\leq
p<+\infty$.
