On the property (db) for continuous function spaces

It is shown that the space C(X) of all continuous real-valued functions with the compact-open topology has the property (db) iff X is a μ-space and a τα-space. As a consequence C(X) has the property (db) iff C(X) is Baire-like answering a question in the literature. Generalizations to the topology of bounded convergence are given.