In this note we continue the study of almost M−open functions between spaces with min-imal structure, also taking into account the unified theory of weakly M−open functions developed by Noiri and Popa. Our main result is a characterization of almost M−open functions via preservation of boundary under inverse image, generalizing a classical characterization of open functions in topological spaces. We partially extend this result to the setting of generalized closure spaces, which allows us to obtain, as a special case, a new characterization of weakly M−open functions in terms of m−θ−boundary preservation.