  We consider regularity for solutions of a class of de Rham's functional
equations. Under some smoothness conditions of functions consisting the
equation, we improve some results in Hata (Japan J. Appl. Math. 1985). Our
results are applicable to some cases that the functions consisting the equation
are non-linear functions on an interval, specifically, polynomials and linear
fractional transformations. Our results imply singularity of some well-known
singular functions, in particular, Minkowski's question-mark function, and,
some small perturbed functions of the singular functions.

Classical Analysis and ODEs;Dynamical Systems