It's known that $\sin(x^2)$ is not periodic.
Are there any theorems which generalise it?
I mean something like "$f(x^n)$ is not periodic if $f$ is a nonconstant periodic function and $n>1$"?
The above conjecture turned out to be false. Dave L. Renfro gave a counterexample.
What if we add continuity assumption?
New conjecture:
If $f$ is a nonconstant continuous periodic function and $n>1$ is an integer, then $f(x^n)$ is not periodic.
On
Your conjecture is not true. Define the function $f$ by letting $f(x) = 0$ if $x$ is transcendental and $f(x) = 1$ if $x$ is algebraic. Then $f$ is periodic (any positive algebraic number is a period), and for each positive integer $n$ the function $g_n$ defined by $g_n(x) = f(x^n)$ is also periodic (any positive algebraic number is a period).
That is correct.
Let $f$ be continuous and $p>0$ a period of $f$. Then $g(x):=f(x^n)$ is also continuous. Assume $q>0$ is a period of it.
The sequence $x_k=\sqrt[n]{kp}$ tends to $+\infty$ while at the same time $x_{k}-x_{k-1}\to 0$. Now fix $x\in\Bbb R$ and let $\delta>0$, where wlog. $\delta<q$. There is $k_0$ such that $x_{k-1}<x_k<x_{k-1}+\delta$ for all $k>k_0$. There is $m$ with $x+mq>x_{k_0}$. Let $k_1$ be minimal with $x_{k_1}>x+mq$. Then $x_{k_1-1}-mq<x<x_{k_1}-mq$ and $(x_{k_1}-mq)-(x_{k_1-1}-mq)<\delta$. Thus $g(y)=f(0)$ for some $y$ with $|y-x|<\delta$. As $\delta$ was arbitrary, we conclude from the continuity of $g$ that $g(x)=f(0)$. Thus $g$ is constant (and in fact so was $f$)