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.
The statement is false.
A counterexample is $$f(x) = \begin{cases} 0 & \text{ if } x \leq 0 \\ e^{-1/x^{2}} & \text { if } x > 0. \end{cases} $$