additive function
Definition 1.
Let $f\mathrm{:}V\mathrm{\to}\mathrm{R}$ be a function^{} on a real vector space $V$ (more generally we can consider a vector space^{} $V$ over a field $F$). We say that $f$ is additive if
$$f(x+y)=f(x)+f(y)$$ 
for all $x\mathrm{,}y\mathrm{\in}V$.
If $f$ is additive, we find that

1.
$f(0)=0$. In fact $f(0)=f(0+0)=f(0)+f(0)=2f(0)$.

2.
$f(nx)=nf(x)$ for $n\in \mathbb{N}$. In fact $f(nx)=f(x)+\mathrm{\cdots}+f(x)=nf(x)$.

3.
$f(nx)=nf(x)$ for $n\in \mathbb{Z}$. In fact $0=f(0)=f(x+(x))=f(x)+f(x)$ so that $f(x)=f(x)$ and hence $f(nx)=f(nx)=nf(x)$.

4.
$f(qx)=qf(x)$ for $q\in \mathbb{Q}$. In fact $qf(px/q)=f(q(px/q))=f(px)=pf(x)$ so that $f(px/q)=pf(x)/q$.
This means that $f$ is $\mathbb{Q}$ linear. Quite surprisingly it is possible to show that there exist additive functions^{} which are not linear (for example when $V$ is a vector space over the field $\mathbb{R}$).
Title  additive function 

Canonical name  AdditiveFunction 
Date of creation  20130322 16:17:31 
Last modified on  20130322 16:17:31 
Owner  paolini (1187) 
Last modified by  paolini (1187) 
Numerical id  9 
Author  paolini (1187) 
Entry type  Definition 
Classification  msc 15A04 
Related topic  LinearFunctional 