If the coordinate functions of $f:D\subset\mathbb{R}\rightarrow\mathbb{R}^{2}$ are periodic, is $f$ periodic too?

Question

I need help with this problem:

Is it true that if the coordinate functions of $f:D\subset\mathbb{R}\rightarrow\mathbb{R}^{2}$ are periodic, then $f$ is periodic?

I think that he answer is yes, since I've done some problems where every time the coordinate functions where periodic, the function was periodic. Am I right? How do I show this or write this correctly? Thanks

Answer 1

No. Suppose that the component functions $f_1, f_2$ of $f : \Bbb R \to \Bbb R^2$ have respective minimal periods $T_1, T_2 > 0$. If $f$ is periodic, say, if $f(x + S) = f(x)$ for some $S > 0$, then $f_1(x + S) = f_1(x)$ and $f_2(x + S) = f_2(x)$. By minimality $S$ is an integer multiple of both $T_1$ and $T_2$ and thus $T_2 / T_1$ is rational.

Contrapositively, if we take functions $f_1, f_2$ with minimal periods $T_1, T_2$ for which $T_2 / T_1$ is irrational, the function $f = (f_1, f_2)$ will not be periodic. Of course, we can construct such a pair by taking a periodic function $f_1$ with minimal period $T_1 > 0$ and declare $f_2(x) := f_1(a x)$ for any irrational $a$.

Answer 2

$f(x)=(\sin \,x , \sin (\sqrt 2 x)$ is easily seen to be a counterexample.