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.

Solution 1:

This has the air of exploiting a hole left in the question parameters, but here comes.

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 $\le 1$),
• so there is no smallest period.