$f:\Bbb R\to \Bbb R$ is continous and nonconstant function. Let $p$ be a positive real number such that $f(x+p)=f(x) $ for all $x \in \Bbb R$ . Then there exitst $n\in\Bbb N$ such that ${p \over n }=min\{a>0|f(x+a)=f(x), \forall x \in \Bbb R\}$.
Is this statement is true?
Sorry, don't have enough points for a comment:
Note that $\frac {p}{n}+\frac{p}{n}+....+\frac{p}{n}$ ($n$ times)=$p$
If $p$ is the smallest period, then $n=1$ . If not, there is a smaller period $p'$. But, since $p$ itself is a period, then there is an integer $r$ with $rp$= $p+p+...+p$ ($r$ times) =$p$