# What is the period of $x\sin x$?

What is the period of $$x\sin x$$?

I’m not able to solve after $$(x+t)\sin(x+t)=x\sin x$$

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• @KanavChadha Do you think $x \sin x$ is periodic? – Toby Mak Feb 8 at 13:13
• t =:0 is a solution. – William Elliot Feb 8 at 13:17

This is because this function is not periodic. Suppose in fact that $$(x + t) \sin(x + t) = x \sin(x)$$ for $$t > 0$$. Then $$\forall n \in \mathbb N: \frac{\pi}{2} = \frac{\pi}{2} \sin\left( \frac{\pi}{2} \right) = \left(nt + \frac{\pi}{2}\right) \sin\left( nt + \frac{\pi}{2} \right).$$ Yet either $$t$$ is a fractional multipl of $$\pi$$, leading to an immediate conradiction by preiodicity of $$\sin$$, or it is not, in which case the orbit of the action by $$n$$ is dense in the values o $$\sin$$ (as the corresponding dynamical system is minimal), leading to another contradiction.
Since $$f(x) = x\sin x$$ has $$f((2n + \frac{1}{2})\pi) = (2n + \frac{1}{2})\pi$$ for any integer $$n$$, it is unbounded and thus not periodic.