Let $ g:\mathbb{R} \to \mathbb{R}$ be a continuous function: $g(x) = g(x+1)$. Show that $g$ satisfies the equation:
$g(x)=\frac{1}{4} \left(g(\frac{x}{2})+g(\frac{x+1}{2})\right)$ for all $x$.
Is it enough to first assume that $g$ satisfies the equation and to check that $g(x)=g(x+1)$ if it does?
It is not enough to assume that $g$ satisfies $$g(x)=\frac{1}{4}\left(g(\tfrac{x}{2})+g(\tfrac{x+1}{2})\right),$$ and then check that $g(x)=g(x+1)$. This is the converse of what you are being asked to prove, and it is not true in general.