On quazianalyticity of infinitely differentiable functions on curves

In W. Rudin’s and I. Brune’s works the following problem has been solved: when $f(x)$ belongs to a certain class $C\{M, I\}$ being an analytical function, their superposition belongs to the class. In this paper it has been shown that W. Rudin’s and I. Brune’s results are true also in the case, when the demand of $\Phi(z)$ analyticity is substituted by a weaker condition. The obtained results can be used for investigation of functions of quasianalytical classes on curves.