An alternative equation for generalized monomials

In this paper we consider a generalized monomial or polynomial f : R → R that satisfies the additional equation f ( x ) f ( y ) = 0 for the pairs ( x , y ) ∈ D , where D ⊆ R 2 is given by some algebraic condition. In the particular cases when f is a generalized polynomial and there exist non-constant regular polynomials p and q that fulfill D = { ( p ( t ) , q ( t ) ) | t ∈ R } or f is a generalized monomial and there exists a positive rational m fulfilling D = { ( x , y ) ∈ R 2 | x 2 - m y 2 = 1 } , we prove that f ( x ) = 0 for all x ∈ R .