---
tag: B1T5SQB154
kind: definition
name: Additive function
msc: [39B22, 39B52]
defines: KF719SZD5T
uses: [082GHNPJ53, 0EPAT51DXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Additive_map
---

## Definition

Let $A$ and $B$ be [[abelian groups|082GHNPJ53]], written additively.
A [[function|0EPAT51DXQ]] $f \colon A \to B$ is *additive* if
$$f(x + y) = f(x) + f(y) \quad \text{for all } x, y \in A.$$
The most common case is $A = B = \R$.

## Remarks

Example: $f \colon \R \to \R$, $f(x) = -5x$, is additive, since $-5(x + y) = -5x + (-5y)$.

Non-example: $f(x) = x^2$ is not additive, since $f(1 + 1) = 4$ while $f(1) + f(1) = 2$.

Every additive function satisfies $f(0) = 0$, $f(-x) = -f(x)$ and $f(nx) = n f(x)$ for $n \in \Z$.
Between real vector spaces, an additive function satisfies $f(qx) = q f(x)$ for every rational $q$, so it is $\Q$-linear.
For arbitrary groups, the same condition defines a group homomorphism.