Monotone Additive Function is Linear/Proof 2
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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:
- $\exists a \in \R: \forall x \in \R: \map f x = a x$
That is, $f$ is a linear function.
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.