I'm trying to show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ is linear if and only if $$f(x) + f(x - a - b) = f(x - a) + f(x - b)$$ for all $a, b, x\in\mathbb{R}$.
The forward direction is trivial and I'm having some problems with the backwards direction. I've thought of using the continuous Cauchy functional equation $$f(x + y) = f(x) + f(y) \Longrightarrow f(x) = cx$$ but I'm having trouble with getting the equation into a usable form and also I'm not sure how to show that the function is continuous.
Any help would be appreciated.
Edit Linear in the sense of $ax + b$ for some constants $a, b$.