---
tag: 1Z0105QF2Q
kind: statement
role: theorem
name: Continuous additive functions are linear
msc: [39B22]
uses: [KF719SZD5T, 0X3WGX95FE, 0BSJSX7ME2]
about: [KF719SZD5T]
sources:
  - url: https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation
---

## Statement

Let $f \colon \R \to \R$ be an [[additive function|KF719SZD5T]], that is, $f(x + y) = f(x) + f(y)$ for all $x, y \in \R$.
If $f$ is [[continuous|0X3WGX95FE]], then
$$f(x) = c x \quad \text{for all } x \in \R, \qquad \text{where } c = f(1).$$

## Remarks

Additivity gives $f(nx) = n f(x)$ for integers $n$, and then $f(q) = q f(1)$ for every rational $q = a/b$, since $b f(a/b) = f(a) = a f(1)$.
The rationals are [[dense|0BSJSX7ME2]] in $\R$, so every real $x$ is a limit of rationals $q_k$, and continuity gives $f(x) = \lim f(q_k) = \lim c q_k = c x$.
For example, if a continuous $f$ satisfies $f(x + y) = f(x) + f(y)$ and $f(1) = 2$, then $f(x) = 2x$, so $f(\pi) = 2\pi$.
Continuity cannot be dropped.
A basis of $\R$ as a vector space over $\Q$ exists by the axiom of choice, and with it one builds additive functions that are not of the form $cx$.
Weaker hypotheses suffice, such as continuity at a single point, monotonicity on an interval, or Lebesgue measurability.
The same argument shows that a continuous additive map between real normed spaces is $\R$-linear.

## Explanations

### L1 human

If a function turns sums into sums and has no jumps, it must be a straight line through the origin.
Additivity alone fixes the function on all fractions, and continuity fills in the remaining points.
This is why a rule like "the cost of two orders together is the sum of their costs", together with continuity, forces a fixed unit price.

### L2 human

This is the regular solution of [[Cauchy's functional equation|ZBMEVWN100]], and many other classical functional equations reduce to it.
For instance, a continuous $g \colon \R \to (0, \infty)$ with $g(x + y) = g(x) g(y)$ satisfies the hypothesis after taking logarithms, so $g(x) = a^x$ with $a = g(1)$.
In linear algebra terms, the theorem says that a continuous homomorphism of the group $(\R, +)$ to itself is automatically $\R$-linear.