Definition:Additive Function (Conventional)

Let $$f: S \to S$$ be a mapping on a algebraic structure $$\left({S, +}\right)$$.

Then $$f$$ is an additive function iff it preserves the addition operation:
 * $$\forall x, y \in S: f \left({x + y}\right) = f \left({x}\right) + f \left({y}\right)$$

Examples
In the field of abstract algebra, this operation can be seen to be a endomorphism on $$\left({S, +}\right)$$.

In the field of linear algebra, it can be seen that a linear transformation is additive.

When the domain is the set of real numbers, this is the Cauchy Functional Equation.

Warning
In the field of number theory, an additive function has a completely different defintion.

In the field of measure theory, an additive function refers to a completely different concept.