Homeomorphic changes of variable and Fourier multipliers

We consider the algebras M_p of Fourier multipliers and show that for every bounded continuous function f on R^d there exists a self-homeomorphism h of R^d such that the superposition f∘h is in M_p(R^d) for all p, 1<p<∞. Moreover, under certain assumptions on a family K of continuous functions, one h will suffice for all f∈K. A similar result holds for functions on the torus T^d. This may be contrasted with the known solution of Luzin's problem related to the Wiener algebra.