Given a metric measure space (X; d; μ) and a Banach function space B over X that has absolutely continuous norm, we prove two results regarding the density in the Newtonian space N^ 1,B (X) of the subclasses consisting of bounded functions, respectively of bounded functions supported in closed balls. We do not assume that μ is a doubling measure. If B is rearrangement invariant, (X; d) is proper and the measure μ is non-atomic, it turns out that the class of bounded compactly supported functions from N^ 1,B (X) is dense in N^ 1,B (X).