if $f(x)f(-x) = 0, \forall x \in\mathbb{R}$ can we prove that $f(x)=0, \forall x \in \mathbb{R}$

Question

I know that if $f(x)g(x) = 0$, that doesn't imply $g(x)=0$ or $f(x)=0$ (there many counterexamples). However how can i prove this ?

Answer 1

Let $f(x)=0$ for $x\geq0$ and $f(x)=1$ for all $x<0$.

Answer 2

Take any non-identically-zero function $f$. Let $A=\{x:f(x)\ne 0\}$. Of course, by the assumption $A\ne\varnothing$. Define $g(x)=0$ on $A$ and $g(x)=\text{whatever you want}$ for $x\not\in A$.