An alternative equation for generalized monomials involving measure

In this paper, we consider a generalized monomial $f:\mathbb{R}\to\mathbb{R}$ that satisfies the additional equation $f(x)f(y)=0$ for the pairs $(x,y)\in D$, where $D\subset{\mathbb{R}}^{2}$ has a positive planar Lebesgue measure. We prove that $f(x)=0$ for all $x\in\mathbb{R}$. Using analogous arguments, we establish a related statement about the signs of such functions: if a generalized monomial $f$ of an even degree is non-negative on a measurable subset of reals with positive Lebesgue measure, then $f(x)\geq 0$ for every real number $x$. Finally, we extend our results to almost monomial functions.