Let $f : \mathbb{R} \to \mathbb{R}$ be a Lebesgue measurable function and $D$ be a countable dense subset of $\mathbb{R}$. Suppose that for a.e. $x \in \mathbb{R}$ we have \begin{equation*} f(x + d) = f(x) \qquad \text{for every } d \in D \end{equation*} (notice that, without loss of generality, we can assume that $D$ is a module over the integers).

Does it follow that $f(x) = c$ a.e.?