  We consider the algebras $M_p$ of Fourier multipliers and show that every
bounded continuous function $f$ on $\mathbb R^d$ can be transformed by an
appropriate homeomorphic change of variable into a function that belongs to
$M_p(\mathbb R^d)$ for all $p$, $1<p<\infty$. Moreover, under certain
assumptions on a family $K$ of continuous functions, one change of variable
will suffice for all $f\in K$. A similar result holds for functions on the
torus $\mathbb T^d$. This may be contrasted with the known result on the Wiener
algebra, related to Luzin's rearrangement problem.
