Proving the solution of a functional equation is constant using Kronecker's theorem

I'm working on the following problem:

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function satisfying $f(x) + f(x + \sqrt{2}) = f(x + \sqrt{3})$ for all $x$. Prove that $f$ is constant.

This problem is from a section on Kronecker's theorem, which leads me to believe that the solution might involve showing that $f$ is constant over a dense subset of $\mathbb{R}$, possibly the subgroup $\langle\sqrt{2},\sqrt{3}\rangle$. However, I'm not sure how to proceed with this approach.

Any hints or suggestions on how to approach this problem would be greatly appreciated.Thank you in advance for your help!