Prove or disprove that $\sin\lfloor x\rfloor$ is periodic.

Question

The title says it all. I was plotting random functions on my phone and noticed this graph. I don't think this function is periodic (WA also agrees). Is there a way to prove if a function like this is periodic or not?

Answer 1

Suppose $f(x+n)=f(x)$ for all $x$.
The discontinuities are at integers for $f(x)$, so must be for $f(x+n)$, so $n$ is an integer.
$\sin n=f(n)=f(0)=0$ so $n$ is a multiple of $\pi$, $n=m\pi$.
$\pi$ is irrational, so $n=0$.
So $f(x)$ is not periodic.