Differentiability of Function with Translation Property

Theorem
Let $f$ be a real function.

Let $f$ have the translation property.

Let $c$ be a real number.

Let $\map {f'} c$ exist.

Then:
 * $f'$ exists


 * $f'$ is a constant function

Proof
Let $x$ be a real number.

We have:

We conclude that $\map {f'} x$ exists as $\map {f'} c$ exists.

Also, $\map {f'} x$ exists everywhere as $x$ is arbitrary.

In other words, $f'$ exists.

Also, $f'$ is a constant function as $\map {f'} x$ equals $\map {f'} c$ for every real number $x$.