= Periodic function without a least positive period
{title2=$f=\mathbf 1_{\mathbb Q},\quad\operatorname{Per}(f)=\mathbb Q$}
Define the <indicator function> of the <rational numbers> on the <real line> by $f(x)=1$ for rational $x$ and $f(x)=0$ otherwise. Every rational shift preserves rationality and irrationality, so every nonzero rational number is a <period of a function>. Conversely, an irrational shift sends $x=0$ from value one to value zero, so it is not a period. The <period group of a function> is exactly $\mathbb Q$, and the positive periods $1/n$ approach zero without a smallest one. The <function> is <nonconstant> and discontinuous everywhere; this is why <continuity> matters in <continuous periodic functions have a least positive period>.
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