On Changes of Variable that Preserve the Absolute Convergence of Fourier–Haar Series of Continuous Functions

It is known that, among all the differentiable homeomorphic changes of variable, only the functions $\varphi_1 (x)=x$ and $\varphi_2 (x)=1-x$, $x\in[0,1]$, preserve the absolute convergence of Fourier–Haar series everywhere. It is established that the class of all differentiable homeomorphic changes of variable that preserve absolute convergence everywhere will not become wider if we restrict ourselves to continuous external functions.