We first briefly expose some crucial phases in studying the space CD₀(Q) = C(Q) + c₀(Q) whose elements are the sums of continuous and “discrete” functions defined on a compact Hausdorff space Q without isolated points. (In this part, special emphasis is on describing the compact space Q* representing the Banach lattice CD₀(Q) as C(Q*).) The rest of the article is dedicated to the analogous frame related to the space CD₀(Q,X) of “continuous-discrete” sections of a Banach bundle X and the space of CD₀-homomorphisms of Banach bundles.

Keywords:Banach lattice, AM-space, Alexandroff duplicate, continuous Banach bundle, section of a Banach bundle, Banach C(Q)-module, homomorphism of Banach bundles, homomorphism of C(Q)-modules.