Definition:Non-Vanishing

Definition
A function $$f$$ is said to be nonvanishing if it has no zeroes in its domain.

That is, $$f$$ is nonvanishing iff:
 * $$\forall x \in \operatorname{Dom} \left({f}\right): f \left({x}\right) \ne 0$$

In this context, $$f$$ is (usually) either real-valued or complex-valued.

In any case, its range needs to contain a zero, so at the very least its range needs to be a ring.