# Must a continuous, non-constant, and periodic functions have a smallest period?

Let $$D\subset\mathbb R$$ and let $$T\in(0,\infty)$$. A function $$f\colon D\longrightarrow\mathbb R$$ is called a periodic function with period $$T$$ if, for each $$x\in D$$, $$x+T\in D$$ and $$f(x+T)=f(x)$$.

If $$D\subset\mathbb R$$ and if $$f\colon D\longrightarrow\mathbb R$$ is a continuous, not constant and periodic function, must there be, among all periods of $$f$$, a minimal one?

I posted a similar question a year ago. The difference is that now I am adding an extra hypothesis, namely that $$f$$ is not constant. All that I was able to prove was that the infimum of the set of periods has to be greater than $$0$$.

• But isn't it true that the infimum of periods must also be a period? If there exists a sequence $(\varepsilon_n)_{n\in \mathbb{N}}$ with $\varepsilon_n\to 0$ such that $p+\varepsilon_n$ is a period of $f$, then, by continuity, for any $x,$ $f(x+p)=\lim_{n\to\infty} f(\varepsilon_n+p+x)=f(x).$ So, if you can prove that the infimum of the set of periods is strictly greater than $0$, aren't you done? Oct 16, 2019 at 8:03
• @WoolierThanThou It seems that the OP is not assuming any topological properties for $D$. If $D$ is not closed, it may be that $x+p\notin D$.
– bof
Oct 16, 2019 at 8:13