  This paper mainly concerns the KAM persistence of the mapping
$\mathscr{F}:\mathbb{T}^{n}\times E\rightarrow \mathbb{T}^{n}\times
\mathbb{R}^{n}$ with intersection property, where $E\subset \mathbb{R}^{n}$ is
a connected closed bounded domain with interior points. By assuming that the
frequency mapping satisfies certain topological degree condition and weak
convexity condition, we prove some Moser type results about the invariant torus
of mapping $\mathscr{F}$ with frequency-preserving under small perturbations.
To our knowledge, this is the first approach to Moser's theorem with
frequency-preserving. Moreover, given perturbed mappings over $ \mathbb{T}^n $,
it is shown that such persistence still holds when the frequency mapping and
perturbations are only continuous about parameter beyond Lipschitz or even
H\"older type. We also touch the parameter without dimension limitation problem
under such settings.
