Assume f is a bounded continuous function on $\mathbf{R}$ and $X$ is a random variable with distribution $F$. Assume for all $x \in \mathbf{R}$ that $$ f(x) = \int_\mathbf{R}f(x+y)F(dy) $$
Please help conclude that $f(x+s) = f(x)$ where $s$ is any value in the support of $F$. The hints that I have come across are to use Martingale theory and consider $\{ X_n\}$ to be i.i.d. with distribution $F$ and make a martingale with some function of $S_n = \sum_{j=1}^nX_j$.
Thanks!