Linearity of Function defined using Function with Translation Property/Lemma

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:

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

Proof

Let $n$ be an integer.

We have:

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

\(=\)

\(\ds \map f {x + n l} - \map f x\)

Definition of $f_x$

\(\ds \)

\(=\)

\(\ds -\paren {\map f x - \map f {x + n l} }\)

\(\ds \)

\(=\)

\(\ds -\paren {\sum_{k \mathop = 1}^n \map f {x + \paren {k - 1} l} - \map f {x + \paren {x + k l} } }\)

Telescoping Series

\(\ds \)

\(=\)

\(\ds -\paren {\sum_{k \mathop = 1}^n \map f x - \map f {x + l} }\)

Definition of Translation Property

\(\ds \)

\(=\)

\(\ds -n \paren {\map f x - \map f {x + l} }\)

\(\ds \)

\(=\)

\(\ds n \paren {\map f {x + l} - \map f x}\)

\(\ds \)

\(=\)

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

Definition of $f_x$

$\blacksquare$