= Continuous periodic functions have a least positive period
{title2=$f\in C(\mathbb R),\ f\text{ nonconstant and periodic}\ \Longrightarrow\ \operatorname{Per}(f)=T_0\mathbb Z$}
For a real-valued <continuous function>, the <period group of a function> is a <closed subgroup>: if periods $T_n$ tend to $T$, <continuity> passes $f(x+T_n)=f(x)$ to the limit. If positive periods could tend to zero, integer multiples of those periods would approach any prescribed displacement $y-x$, forcing $f(y)=f(x)$. A <nonconstant function> therefore has a positive lower bound for its positive periods. Their <infimum> $T_0$ is itself a period by closedness. Subtracting an integer multiple of $T_0$ from any other period leaves a period in $[0,T_0)$, which must be zero. This proves the stated structure without assuming <differentiability>.
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