Let $~f : E \rightarrow \mathbb{R}, ~ E \subset \mathbb{R}~~$ be a periodic fucntion and $ S = \{ T \in \mathbb{R} ~ : ~ \forall x\in \mathbb{R} ~~ f(x+T) = f(x) \} $ be the set of all periods of fucntion f.
We know that if $ ~T~ $ is a period of fucntion $~f~$ then $ \forall n \in \mathbb{N} ~~ nT ~$ is also a period, so the set S is at least countably infinite. We also know that if any real number is a period of $ ~f~ $ than $~ f$ is constant, so for constant fucntion the set of all periods is uncountable.
I have the following question: if the set S of all periods of function $~f~$ is uncountable does it mean that $~f~$ is a constant? Or there is an example of non-constant periodic function which has this property?
Thanks in advance!
The set of periods of a function $f\colon \mathbb{R}\to\mathbb{R}$ can be any subgroup of the additive real numbers. Indeed, if $S$ is any subgroup of $\mathbb{R}$, then any one-to-one function $\overline{f}\colon\mathbb{R}/S \to \mathbb{R}$ lifts to a periodic function $f\colon\mathbb{R}\to\mathbb{R}$ whose periods are precisely the elements of $S$.
There are a wide variety of different subgroups of the real numbers. For example, the real numbers are a vector space over $\mathbb{Q}$ with uncountable dimension, and any vector subspace is a subgroup. In particular, $\mathbb{R}$ has plenty of proper uncountable subgroups.