I am confused by a discussion with a colleague. The discussion is about the period of a periodic function.
For example, the periodic function $$f(x)=\sin(x), \quad x\in (0,\infty)$$ has period $2\pi$. If I change the scale and build the function, $$g(x)=\sin(\ln x),\quad x\in (0,\infty)$$ is this new function, g, periodic? If it is, what is the period?
EDIT
I will clarify my point. If I change the scale of the function $g$, let's say, $\ln x =u$ then I will have function $$h(u)=\sin u, \quad u\in \mathbb R$$ and now $h$ is periodic on $u\in \mathbb R $.
So, my point is can I say that $g$ is not periodic in $x$-domain but it is in $\log$-domain?
Best Answer
$g$ is not periodic as the difference between two consecutive roots is unbounded as we consider the roots going to $\infty$.