# Must a continuous 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 $$f\colon D\longrightarrow\mathbb R$$ is continuous and periodic, must there be, among all periods of $$f$$, a minimal one?

Questions like this one have been posted here before, but in each case, as far as I can see, the domain of $$f$$ was $$\mathbb R$$, which implies that the set $$P$$ of periods, together with $$0$$ and $$-P$$, is a subgroup of $$(\mathbb{R},+)$$. Using that (together with continuity), it is easy to see that a minimal period must exist indeed. But I don't know whether it is true or not in the general case.

• Maybe I misunderstand, but I feel like the argument carries over pretty cleanly. Suppose there is no minimal one. Because periods get arbitrarily small, $D$ must be dense in a positive ray (starting at any $x \in D$), and in fact $f$ is constant on this dense set. Thus $f$ is constant. – Mees de Vries Oct 11 '18 at 10:37
• @MeesdeVries How do you go from “periods get arbitrarily small” to “$D$ must be dense in a positive ray”? Doesn't that assume that if $T$ and $T^\star$ are periods, with $T>T^\star$, then $T-T^\star$ is a period too? Is that obvious in this context? – José Carlos Santos Oct 11 '18 at 10:40
• If $T_n$ is a sequence of periods of $f$ which tends to zero, and $x \in D$, then the set of points $x'$ where $f(x) = f(x')$ -- so in particular $x' \in D$ -- includes at least $\{x + kT_n \mid k, n \in \mathbb N\}$, which is certainly dense in $[x, \infty)$. Is this wrong? – Mees de Vries Oct 11 '18 at 10:44
• @MeesdeVries It looks right, but you are assuming that “there is no minimal period” is equivalent to “there is a sequence of periods which converges to $0$”. Why do you think so? – José Carlos Santos Oct 11 '18 at 10:47
• Ah, thank you for the correction. I see where the difficulty lies. Interesting question! – Mees de Vries Oct 11 '18 at 10:48

Let $$D=\{0\}\cup(1,\infty)$$ and let $$f(x)$$ be a constant function. Then $$T$$ will be a period if and only if $$T+D\subseteq D$$. In particular:
• every $$T>1$$ is a period, but
• $$T=1$$ is not a period (and there cannot be smaller periods $$\le 1$$),