# a Function with several periods

A periodic function is given by $f(x+nT)=f(x)$, with 'n' an integer and T the period.

My question is if we can define a non-constant function with several periods; by that, I mean

$f(x+T_{i})=f(x)$ with $i=1,2,3,4,\dots$ a set of different numbers.

For example, a function that satisfies $f(x+2)=f(x)$, as well as $f(x+5.6)=f(x)$ and $f(x+ \sqrt 2) =f(x)$, but $f(x)$ is NOT a constant.

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If we work with functions from the real line to itself, we have several cases. Since the set of periods of a function is a subgroup of the additive group of the real, it is either dense in $\Bbb R$ (for the usual topology) or discrete (of the form $a\Bbb Z$ for some $a\in\Bbb R$.
If $G$ is a subgroup of $\Bbb R$, then the characteristic function of $G$ is $g$-periodic for each $g\in G$.
The group of the periods of a continuous function is either $\Bbb R$ (constant functions) or discrete.