How can I prove that this function is periodic :

Question

I know that a periodic function satisfies $F(x+a)=F(x)$

$y=\left\lfloor\frac{\sin x}x\right\rfloor$

And the period of this fuction is $2\pi$ But putting $f(x+2\pi)$ I can't evaluate denominator inside the greatest integer function
So how can I prove this function periodic?

Answer 1

For any single variable function to be periodic,

$$ f(x) = f(x+ \lambda)$$

should be strictly satisfied. Just having (at least) two sequence of its roots in arithmetic progression is not enough.

Answer 2

The result is wrong

First of all you have an issue to deal with at $0$.

Second $y$ is equal to zero on $I=(-\pi, \pi)\setminus \{0\}$. And there is no other translated subset of $I$ in $\mathbb R$ on which $y$ takes such value.