# Definition:Additive Function (Conventional)

*This page is about functions preserving addition. For other uses, see Definition:Additive Function.*

## Definition

Let $f: S \to S$ be a mapping on an algebraic structure $\struct {S, +}$.

Then $f$ is an **additive function** if and only if it preserves the addition operation:

- $\forall x, y \in S: f \paren {x + y} = f \paren x + f \paren y$

## Examples

In the field of abstract algebra, this operation can be seen to be a endomorphism on $\struct {S, +}$.

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.

## Also see

- Results about
**additive functions**can be found here.