Jensen like: $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2} $for $x,y \in \mathbb{R}$ that $x-y \in \mathbb{Z}$

Question

Solve functional equation, where $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfy $$f(a)=\frac{f(a-\frac{k}{2})+f(a+\frac{k}{2})}{2},$$ for $a\in \mathbb{R}$ and $k\in \mathbb{Z}.$

I obtained this equation from another one. I know that this function $f$ satisfy that $f=g'$ where $g$ is continuous on $\mathbb{R}$. This equation looks similar to Jensen equation, but the arguments differs about integer here. I don't have another idea than using Jensen equation... I would be grateful for help.

Answer 1

The functional equation implies that the graph of $f$ is point symmetric about every point $(a,f(a))$, and hence $f$ is linear.