$f(x)$ is linear if and only if $f(x) + f(x - a - b) = f(x - a) + f(x - b)$ 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 $a, b$.

Proof of cauchy continuity condition:
We have $f(0+0)= f(0)+f(0)$ this gives $f(0)= 0$.
And we can show that $f(-x)=-f(x)$.
For integer value $n$, we can show that $f(nx)= nf(x)$  Also for $n\neq 0$ we can show that $f(\frac{x}{n})= \frac{1}{n}f(x)$
This gives for $m\neq 0$ integer, we have:
$$f(\frac{n}{m}x)=\frac{n}{m}f(x)$$
Now if $x\in \mathbb R$, then there is sequence of continuous function of rational $q_n$ such that $\lim q_n\to x$.
Now $f(x)=\lim f(q_n)$ as $f$ is continuous. Hence
$$f(x)= \lim  q_n f(1)= x.f(1)= cx$$
Where $f(1)=c$
