In the paper, the space CD₀(Q,X) = C(Q,X) + c₀(Q,X) is considered whose elements are the sums of continuous and discrete sections of a Banach bundle X over a compact Hausdorff space Q without isolated points. As is known, this space is isometric to the space C(Q*,X*) of continuous sections of some Banach bundle X* over the compact space Q* = Q x {0,1} (with a special topology). We clarify the connection between X and X* on the example of subbundles as well as bundles obtained by continuous change of variables and by restriction onto a topological subspace. In addition, we introduce and study the space CD₀[X,Y] of CD₀-homomorphisms of Banach bundles X and Y and show that is possesses some properties analogous to those of the space of CD₀-sections.

Keywords:continuous Banach bundle, section of a Banach bundle, Banach C(Q)-module, homomorphism of Banach bundles, homomorphism of C(Q)-modules.