Two periodic functions whose product is not periodic

Question

I'm looking for a concrete example. One example that @Koro suggested in the chat is $\{\sqrt{2}x\}\sin x$ where $\{-\}$ is the fractional part function. Here is a plot of that function. Seems periodic but if you take a closer look, it's not periodic. How can I prove it's not periodic? or is there any simpler example?

Answer 1

You can consider any two functions whose periods are incommensurable, for example $\sin(x)\sin(\pi x)$, or $\{x\}\{\sqrt 2 x\}$.

Edit: On better though, that may be false in general.

But since you are looking for a concrete example, consider this one: $\cos(x) \cos(\pi x)$. Observe that if $\cos(x)\cos(\pi x)=1$ then we have $\cos(x)=1$ and $\cos(\pi x)=1$ or $\cos(x)=-1$ and $\cos(\pi x)=-1$. In the first case, from the first equation we have $x=2\pi k$ for some integer $k$. From the second equation we have $x=2m$ for some integer $m$. So we have $\pi k = m$ for integers $k$ and $m$. If $k\ne 0$ then $\pi = m/k \in \Bbb Q$, a contradiction. So $k=0$ and then $x=0$. A similar argument shows that the second case can't happen.

This proves that $\cos(x)\cos(\pi x)$ take the value $1$ only once, hence cannot be periodic.