Product of functions is $0$ but none of the functions is identically $0$? 
Given that $f(x) \cdot g(x) = 0$ for all $x$ is it true that at least one of the functions is $0$ for all $x$?

The correct answer is this doesn't necessarily hold true. Can you give such example? I have a feeling this had something to do with piecewise functions.

One continuous example: $f(x)=|x|+x, g(x)=|x|-x$.
For any $x$, at least one of the functions must be $0$, but there is nothing stopping them from "sharing" the $x$-axis between them.
