Monotone Additive Function is Linear/Proof 2

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $f: \R \to \R$ be a monotone real function which is additive, that is:

$\forall x, y \in \R: \map f {x + y} = \map f x + \map f y$

Then:

$\exists a \in \R: \forall x \in \R: \map f x = a x$


Proof

We use a Proof by Contraposition.

To that end, suppose $f$ is not linear.

We know that Graph of Nonlinear Additive Function is Dense in the Plane.

Therefore $f$ is not bounded on any nonempty open interval.

But then $f$ is certainly not monotone.

Hence, by Rule of Transposition, if $f$ is monotone, then it is linear.