Let $F:\mathbb R^n\to\mathbb R$ is a non-constant continuous function. Is it true that $Leb[(x_1,...,x_n)\in\mathbb R^n:F(x_1,...,x_n)=0]=0?$ Here $Leb$ denotes Lebesgue measure.
I don't know if this is a well known result. I have heard something like graph of a continuous function has Lebesgue measure 0. Is this related to that? I don't even know how to prove this latter fact so if you can include a proof I would be delighted.