  The classical $C^0$ linearization theorem for the non-autonomous differential
equations states the existence of a $C^0$ topological conjugacy between the
nonlinear system and its linear part. That is, there exists a homeomorphism
(equivalent function) $H$ sending the solutions of the nonlinear system onto
those of its linear part. It is proved in the previous literature that the
equivalent function $H$ and its inverse $G=H^{-1}$ are both H\"{o}lder
continuous if the nonlinear perturbation is Lipschitzian. Questions: is it
possible to improve the regularity? Is the regularity sharp? To answer this
question, we construct a counterexample to show that the equivalent function
$H$ is exactly Lipschitzian, but the inverse $G=H^{-1}$ is merely H\"{o}lder
continuous. Furthermore, we propose a conjecture that such regularity of the
homeomorphisms is sharp (it could not be improved anymore). We prove that the
conjecture is true for the systems with linear contraction. Furthermore, we
present the special cases of linear perturbation, which are closely related to
the spectrum.
