## definition

A function is called additive if it has the functional equation${\ displaystyle f}$

${\ displaystyle f (x + y) = f (x) + f (y)}$

Fulfills. If the definition and target area are Abelian groups , one also speaks of - linearity . ${\ displaystyle \ mathbb {Z}}$

## Sub and super additive functions

If there is a semigroup with the link , then a mapping is called subadditive if the following applies to all and from : ${\ displaystyle M}$ ${\ displaystyle +}$${\ displaystyle f \ colon M \ to \ mathbb {R}}$ ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle M}$

${\ displaystyle f (x + y) \ leq f (x) + f (y)}$.

The figure is called superadditive if the following applies to all and off : ${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle M}$

${\ displaystyle f (x + y) \ geq f (x) + f (y)}$.

## properties

• If it is an additive function, then for every finite number of elements :${\ displaystyle f}$${\ displaystyle x_ {1}, \ dotsc, x_ {n}}$${\ displaystyle M}$
${\ displaystyle f (x_ {1} + \ dotsb + x_ {n}) = f (x_ {1}) + \ dotsb + f (x_ {n})}$
The same applies to sub- and superadditivity.

## Definition in number theory

With number theoretic functions one regards a link to the multiplication. A number theoretic function is called additive if the equation ${\ displaystyle f \ colon \ mathbb {N} \ to \ mathbb {C}}$${\ displaystyle \ mathbb {N}}$

${\ displaystyle f (xy) = f (x) + f (y)}$

for all coprime and applies. If this even applies to all and , the function is called strictly additive . ${\ displaystyle x}$${\ displaystyle y \ in \ mathbb {N}}$${\ displaystyle x}$${\ displaystyle y}$

There is a similar restriction of additivity (to disjoint instead of arbitrary unions) in measure theory.