The question was which of the following is a periodic function, in which two of the possible answers which I thought of were $\sin{\{x\}}$ and $\sin[x]$ where $\{.\}$ and $[.]$ are fractional part and greatest integer functions respectively. The answer is the former one. But my question is why is it not the latter one. I mean $\sin()$ is a periodic function. At some point it will repeat itself the the latter case too.
Please also give the periods of both the functions(or whichever is periodic) please. Thanks in advance.
Best Answer
The period of $\sin(x)$ is $2\pi$.
For $f(x)=\sin\lfloor x\rfloor$, note that $f(0)=0$ only for $x\in[0,1)$. For other values, $f(x) = \sin(n), n\in \Bbb N$. But $\sin(x) = 0 \iff x = k\pi, k \in \Bbb Z$. Thus $f(x)$ is never zero again, or the function is not periodic.