TITLE: Product of functions is $0$ but none of the functions is identically $0$?
QUESTION [11 upvotes]: 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.

REPLY [5 votes]: $$f(x)\cdot g(x)=0\iff f(x)=0\lor g(x)=0.$$
Then take two distinct values $x_1,x_2$ and define the functions in such a way that $f(x_1)\ne0$ and $g(x_2)\ne0$ and zeroes everywhere else.