  Let $\Omega$ be a locally compact Hausdorff space. We show that any local
$\mathbb{C}$-linear map (where "local" is a weaker notion than
$C_0(\Omega)$-linearity) between Banach $C_0(\Omega)$-modules are "nearly
$C_0(\Omega)$-linear" and "nearly bounded". As an application, a local
$\mathbb{C}$-linear map $\theta$ between Hilbert $C_0(\Omega)$-modules is
automatically $C_0(\Omega)$-linear. If, in addition, $\Omega$ contains no
isolated point, then any $C_0(\Omega)$-linear map between Hilbert
$C_0(\Omega)$-modules is automatically bounded. Another application is that if
a sequence of maps $\{\theta_n\}$ between two Banach spaces "preserve
$c_0$-sequences" (or "preserve ultra-$c_0$-sequences"), then $\theta_n$ is
bounded for large enough $n$ and they have a common bound. Moreover, we will
show that if $\theta$ is a bijective "biseparating" linear map from a "full"
essential Banach $C_0(\Omega)$-module $E$ into a "full" Hilbert
$C_0(\Delta)$-module $F$ (where $\Delta$ is another locally compact Hausdorff
space), then $\theta$ is "nearly bounded" (in fact, it is automatically bounded
if $\Delta$ or $\Omega$ contains no isolated point) and there exists a
homeomorphism $\sigma: \Delta \rightarrow \Omega$ such that $\theta(e\cdot
\varphi) = \theta(e)\cdot \varphi\circ \sigma$ ($e\in E, \varphi\in
C_0(\Omega)$).
