  We generalize our recent results for the boundary and interface charges in
one-dimensional single-channel continuum [Phys. Rev. B 104, 155409 (2021)] and
multichannel tight-binding [Phys. Rev. B 104, 125447 (2021)] models to the
realm of the multichannel continuum systems. Using the technique of boundary
Green's functions, we give a rigorous proof that the change in boundary charge
upon the shift of the system towards the boundary by the distance
$x_{\varphi}\in[0, L]$ is given by a perfectly linear function of $x_{\varphi}$
plus an integer-valued topological invariant $I$ -- the so called boundary
invariant. For systems with weak potential amplitudes, we additionally develop
Green's function-based low-energy theory, allowing one to analytically access
the physics of multichannel continuum systems in the low-energy approximation.
