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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).

Vitor Borges
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  • Related: [The sum of two continuous periodic functions is periodic if and only if the ratio of their periods is rational?](http://math.stackexchange.com/questions/1356802/), and [this answer](http://math.stackexchange.com/a/775731) to [A real continuous periodic function with two incommensurate periods is constant](http://math.stackexchange.com/questions/775718/). Note that non-continuous periodic functions do not necessarily have a minimum period, so the answer to your question depends on whether you mean *some* periods $p,q$ or *all* periods $p,q$. – dxiv Feb 21 '17 at 20:50

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