Are there two periodic functions $f,g:\mathbb{R}\to\mathbb{R}$ with period $p,\,q$, respectively, such that $\frac pq \not\in\mathbb{Q}$ and $f+g$ is periodic?
The question arised when I tried to prove that the set of all periodic functions is not a vector subspace of the set of functions from $\mathbb{R}$ to $\mathbb{R}$, which I did by showing that the functions $f(x) = \{x\} = x - \lfloor x\rfloor$ and $g(x) = \sin(\sqrt2x)$ are both periodic funcions whose sum is not.
To do that, I had to use particular properties of such functions and didn't get much far into the general case (supposing that the answer to the question is no).
