Product of Negligible and Non-Negligible Functions

Question

I know that the product of two negligible functions will always be negligible, but I'm wondering if it's possible for the product of two non-negligible functions to be a negligible function?

Answer 1

I'm wondering if it's possible for the product of two non-negligible functions to be a negligible function?

Yes, actually; here is an example:

Consider the two functions:

$$P(x) = 1 \text{ if x is an even integer}, 0 \text{ otherwise}$$ $$Q(x) = 1 \text{ if x is an odd integer}, 0 \text{ otherwise}$$

Both $P$ and $Q$ are nonnegligible functions.

However $P(x)Q(x) = 0$, which is (trivially) a negligible function.