Additive Function is Linear for Rational Factors
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Theorem
Let $f: \R \to \R$ be an additive function.
Then:
- $\forall r \in \Q, x \in \R: \map f {x r} = r \map f x$
Proof
Trivially, we have:
- $\forall x \in \R: \map f {1 \cdot x} = 1 \map f x$
Next, suppose that:
- $\map f {n x} = n \map f x$
By additivity of $f$, we have:
\(\ds \map f {\paren {n + 1} x}\) | \(=\) | \(\ds \map f {n x + x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map f {n x} + \map f x = n \map f x + \map f x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {n + 1} \map f x\) |
Hence by the Principle of Mathematical Induction:
- $\forall n \in \N, x \in \R: \map f {n x} = n \map f x$
As Additive Function is Odd Function and Odd Function of Zero is Zero, we conclude:
- $\forall p \in \Z, x \in \R: \map f {p x} = p \map f x$
Let $p \ne 0$.
By substituting $y = p x$, the above gives:
- $\forall p \in \Z \setminus \set 0, y \in \R: \map f y = p \map f {\dfrac y p}$
In other words:
- $\forall p \in \Z \setminus \set 0, y \in \R: \map f {\dfrac y p} = \dfrac 1 p \map f y$
Given $p, q \in \Z, q \ne 0$, we have:
\(\ds \map f {\dfrac p q} x\) | \(=\) | \(\ds \map f {p \dfrac x q}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p \map f {\dfrac x q}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds p \dfrac {\map f x } q\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac p q \map f x\) |
Therefore we conclude:
- $\forall r \in \Q, x \in \R: \map f {r x} = r \map f x$
$\blacksquare$