Continuous periodic functions have a least positive period ($f\in C(\mathbb{R}),f\text{ nonconstant and periodic}⟹\mathrm{Per}(f)=T_0\mathbb{Z}$)

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.