How to solve the functional equation $f(x+a)=f(x)+a$ Are there any other solutions of the functional equation $f(x+a)=f(x)+a$ ($a=\mathrm{const}$, $a\in\mathbb{R}\setminus\left\{0\right\}$) apart from $f(x)=x+C$ ($C=\mathrm{const}$)?
Edit: $a$ is a fixed number here.

No if the equation holds for all $a \in \mathbb{R} \setminus \{0\}$. The functional equation $f(x + a) = f(x) + a$ implies that
$$\frac{f(x + a) - f(x)}{a} = 1.$$
Taking the limit of both sides as $a \to 0$ shows that $f'$ exists and $f'(x) = 1$, so the only solutions are $f(x) = x + C$. as you stated.
