Linearity of Function defined using Function with Translation Property

Theorem

Let $f$ be a real function.

Let $f$ have the translation property.

Let $x$ and $l$ be real numbers.

Define:

$\map {f_x} l = \map f {x + l} - \map f x$

Then:

$\forall q \in \Q: \map {f_x} {q l} = q \map {f_x} l$

Proof

Lemma

Let $f$ be a real function.

Let $f$ have the translation property.

Let $x$ and $l$ be real numbers.

Define:

$\map {f_x} l = \map f {x + l} - \map f x$

Then:

$\forall n \in \Z: \map {f_x} {n l} = n \map {f_x} l$

$\Box$

Let $q$ be a rational number.

Choose integers $n$, $m$ such that:

$\dfrac n m = q$

We need to prove that:

$\map {f_x} {q l} = q \map {f_x} l$

We have:

\(\ds \map {f_x} {q l}\)

\(=\)

\(\ds \map {f_x} {\frac n m l}\)

Definition of $n$, $m$

\(\ds \)

\(=\)

\(\ds n \map {f_x} {\frac l m}\)

Lemma

\(\ds \)

\(=\)

\(\ds \frac n m m \map {f_x} {\frac l m}\)

\(\ds \)

\(=\)

\(\ds \frac n m \map {f_x} {\frac {m l} m}\)

Lemma

\(\ds \)

\(=\)

\(\ds q \map {f_x} l\)

Definition of $n$, $m$

$\blacksquare$