If $f\left(x+\frac{5}{6}\right)+f(x)=f\left(x+\frac{1}{2}\right)+f\left(x+\frac{1}{3}\right)$ then check for periodicity

Question

Let $f:\mathbb R\rightarrow \mathbb R$ be a function which satisfies the functional equation $$f\left(x+\frac{5}{6}\right)+f(x)=f\left(x+\frac{1}{2}\right)+f\left(x+\frac{1}{3}\right)$$ Is the function periodic ?

I tried by replacing $x$ by $x+\frac{1}{2}$ and also $x+\frac{1}{3}$ and so on but couldn't reach any conclusion

Answer 1

No, it does not have to be periodic. A linear function $f(x)=ax$ satisfies this relation, as you can easily check.