## 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:

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

## 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.

### Example: $\map f x = 3 x$

Let $\map f x$ be the real function defined as:

$\forall x \in \R: \map f x = 3 x$

Then $f$ is an additive function.

### Square Root is not Additive

The square root function is not an additive function.