Prove that $\sin(\sqrt x)$ not periodic

Question

$\sin\sqrt x$ is not a periodic function. How can one prove this?

Answer 1

An other way to prove it is to remark that if this function is $T-$ periodic, the derivate would be $T-$ periodic too. Indeed,

$$f'(x+T)=\lim_{h\to 0}\frac{f(x+T+h)-f(x+T)}{h}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=f'(x).$$

Therefore if $f(x)=\sin(\sqrt{x})$, then $f'(x)=\frac{\cos(\sqrt x)}{2\sqrt x}$ would be periodic, and this is impossible because $$\lim_{x\to \infty }f'(x)=0.$$

Answer 2

Assume that it is. Then $\sin{\sqrt{x}}=\sin{\sqrt{x+p}}=$ for all $x$ and some $p$. Solve for $p$.