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 codomain needs to contain a zero, so at the very least its codomain needs to be a ring.