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?
Two periodic functions whose product is not periodic
analysisperiodic functions
Best Answer
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.