# Definition:Additive Function (Conventional)

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

## Contents

## 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.

## Also see

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

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**additive**:**1.**(of a function between semigroups) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**additive function**