Difference and functional equations Definition

Additive function

Definition

Let $A$ and $B$ be abelian groups, written additively. A function $f \colon A \to B$ 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.

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Abelian group concept
Function concept
Additive function definition

Tag	`B1T5SQB154`
URI	`m2://B1T5SQB154`
Kind	definition
Writing	W0 stub
MSC	39B22, 39B52
Defines	Additive function
Uses	Abelian group, Function

Additive function

Definition

Remarks

Reading path

Further reading

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