  In this paper, we study the persistence and remaining regularity of KAM
invariant torus under sufficiently small perturbations of a Hamiltonian
function together with its derivatives, in sense of finite smoothness with
modulus of continuity, as a generalization of classical H\"{o}lder continuous
circumstances. To achieve this goal, we extend the Jackson approximation
theorem to the case of modulus of continuity, and establish a corresponding
regularity theorem adapting to the new iterative scheme. Via these tools, we
establish a KAM theorem with sharp differentiability hypotheses, which asserts
that the persistent torus keeps prescribed universal Diophantine frequency
unchanged and reaches the regularity for persistent KAM torus beyond H\"older's
type.
