I encountered in my research a functional equation that I'm not sure how to solve in general. It is similar to Cauchy's functional equation but includes an extra constant term of $-f(0)$. I'm not an expert in functional equations, so any help would be appreciated.
The functional equation is $f(x+y)=f(x)+f(y)-f(0)$
This is actually a part of a pair of functional equations that $f$ has to satisfy, but I'm at first interested in just solving this first functional equation. The other equation is $f(-x)=-f(x)+2f(0)$, and $f$ has to satisfy both the above one and this other one.
Clearly $f(x) = x$ solves the functional equations, as does any function of the form $f(x)=ax+b$, but are there any other solutions?
Any help is much appreciated.