If $f$ is constant function then every real number $p>0$ is its period.
I was wondering is the converse true, that is:
If every real number $p>0$ is period of a function $f : \mathbb R \to \mathbb R $, is $f$ then a constant function?
I believe that the answer is yes if we add additional condition that $f$ is continuous but I want to know is this true without the requirement of continuity, in other words, are there discontinuous functions which also have every real number $p>0$ as its period?
It works without continuity : assume that $f:\mathbb{R}\to\mathbb{R}$ is a such function. Let $k=f(0).$ Then for $x\in\mathbb{R}^*,$ because $f$ is $x-$periodic by hypothesis, you get $$f(x)=f(0+x)=f(0)=k$$ and then $f$ is constant.