Separately nowhere constant functions; n-cube and \alpha-prism densities

A function f from a countable product X of of Polish spaces Xi into a Polish space is separately nowhere constant provided it is nowhere constant on every section of X. We show that every continuous separately nowhere constant function is one-to-one on a product of perfect subsets of Xi\u27s. This result is used to distinguish between n-cube density notions for different n\leq\omega, where \omega-cube density is a basic notion behind the Covering Property Axiom CPA formulated by Ciesielski and Pawlikowski. We will also distinguish, for different values of \alpha\u3c\omega1, between the notions of \alpha-prism densities --- the more refined density notions used also in CPA